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10.1.1: Solving One-Step Equations Using Properties of Equality - Mathematics


Learning Objectives

  • Solve algebraic equations using the addition property of equality.
  • Solve algebraic equations using the multiplication property of equality.

Writing and solving equations is an important part of mathematics. Algebraic equations can help you model situations and solve problems in which quantities are unknown. The simplest type of algebraic equation is a linear equation that has just one variable.

An equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side. Expressions are made up of terms, and the number of terms in each expression in an equation may vary.

Algebraic equations contain variables, symbols that stand for an unknown quantity. Variables are often represented with letters, like ( x), ( y), or ( z). Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of ( 3x) is 3.

An important property of equations is one that states that you can add the same quantity to both sides of an equation and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation “balanced.” If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.

Let’s look at a simple numeric equation, ( 3+7=10), to explore the idea of an equation as being balanced.

The expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality. Let’s see what happens when 5 is added to each side.

( 3+7+5=10+5)

Since each expression is equal to 15, you can see that adding 5 to each side of the original equation resulted in a true equation. The equation is still “balanced.”

On the other hand, let’s look at what would happen if you added 5 to only one side of the equation.

( egin{array}{c}
3+7=10 \
3+7+5=10 \
15 eq 10
end{array})

Adding 5 to only one side of the equation resulted in an equation that is false. The equation is no longer “balanced”, and it is no longer a true equation!

Addition Property of Equality

For all real numbers ( a), ( b), and ( c): If ( a=b), then ( a+c=b+c).

If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.

When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.

When the equation involves addition or subtraction, use the inverse operation to “undo” the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to 0, the additive identity.

Example

Solve ( x-6=8).

Solution

( x-6=8)This equation means that if you begin with some unknown number, ( x), and subtract 6, you will end up with 8. You are trying to figure out the value of the variable ( x).
( egin{array}{rr}
x-6&= 8
+6 & +6 \
hline x+0&= 14
end{array})

Using the Addition Property of Equality, add 6 to both sides of the equation to isolate the variable. You choose to add 6, as 6 is being subtracted from the variable.

Subtracting 6 from both sides leaves you with ( x+0=14).

( x=14)

Since subtraction can be written as addition (adding the opposite), the addition property of equality can be used for subtraction as well. So just as you can add the same value to each side of an equation without changing the meaning of the equation, you can subtract the same value from each side of an equation.

Example

Solve ( x+7=42).

Solution

( x+7=42)Since 7 is being added to the variable, subtract 7 to isolate the variable.
( egin{array}{rr}
x+7= & 42
-7 & -7 \
hline x+0= & 35
end{array})

To keep the equation balanced, subtract 7 from both sides of the equation.

This gives you ( x+0=35).

( x=35)

Advanced Example

Solve ( 12.5+x=-7.5).

Solution

( 12.5+x=-7.5)Since 12.5 is being added to the variable, subtract 12.5 to isolate the variable.
( egin{array}{rr}
12.5+x= & -7.5
-12.5 & -12.5 \
hline 0+x= & -20
end{array})

To keep the equation balanced, subtract 12.5 from both sides of the equation.

This gives you ( 0+x=-20).

( x=-20)

The examples above are sometimes called one-step equations because they require only one step to solve. In these examples, you either added or subtracted a constant from both sides of the equation to isolate the variable and solve the equation.

Exercise

What would you do to isolate the variable in the equation below, using only one step?

( x+10=65)

  1. Add 10 to both sides of the equation.
  2. Subtract 10 from the left side of the equation only.
  3. Add 65 to both sides of the equation.
  4. Subtract 10 from both sides of the equation.
Answer
  1. Add 10 to both sides of the equation.

    Incorrect. Adding 10 to both sides of the equation gives an equivalent equation, ( x+20=65+10), but this step does not get the variable alone on one side of the equation. The correct answer is: Subtract 10 from both sides of the equation.

  2. Subtract 10 from the left side of the equation only.

    Incorrect. Subtracting 10 from the left side will isolate the variable, but subtracting 10 from only one side of the equation does not keep the equation balanced. According to the properties of equality, you must perform the same exact operation to each side of the equation, so you must also subtract 10 from 65 to keep the equation balanced. The correct answer is: Subtract 10 from both sides of the equation.

  3. Add 65 to both sides of the equation.

    Incorrect. This step will not isolate the variable. It will only give an equivalent equation. ( x+10+65=65+65). The correct answer is: Subtract 10 from both sides of the equation.

  4. Subtract 10 from both sides of the equation.

    Correct. Subtracting 10 from each side of the equation yields an equivalent equation with the variable isolated to give the solution: ( x+10-10=65-10), so ( x=55).

Advanced Question

What would you do to isolate the variable in the equation below, using only one step? ( x-frac{1}{4}=frac{7}{2})

  1. Subtract ( frac{1}{4}) from both sides of the equation.
  2. Add ( frac{1}{4}) to both sides of the equation.
  3. Subtract ( frac{7}{2}) from both sides of the equation.
  4. Add ( frac{7}{2}) to both sides of the equation.
Answer
  1. Subtract ( frac{1}{4}) from both sides of the equation.

    Incorrect. Subtracting ( from both sides of the equation gives the equation,frac{1}{4}) from both sides of the equation gives the equation, ( x-frac{1}{4}-frac{1}{4}=frac{7}{2}-frac{1}{4}), which is the same as ( x-frac{1}{4}-frac{1}{4}=frac{14}{4}-frac{1}{4}) or ( x-frac{1}{2}=frac{13}{4}). However, this step does not get the variable alone on one side of the equation. The correct answer is: Add ( frac{1}{4}) to both sides of the equation.

  2. Add ( frac{1}{4}) to both sides of the equation.

    Correct. Adding ( frac{1}{4}) to each side of the equation yields an equivalent equation and isolates the variable: ( x-frac{1}{4}+frac{1}{4}=frac{7}{2}+frac{1}{4}) and ( x=frac{14}{4}+frac{1}{4}), so ( x=frac{15}{4}).

  3. Subtract ( frac{7}{2}) from both sides of the equation.

    Incorrect. Subtracting ( frac{7}{2}) from both sides will result in the equivalent expression ( x-frac{1}{4}-frac{7}{2}=frac{7}{2}-frac{7}{2}), which can be rewritten ( x-frac{1}{4}-frac{14}{4}=0), or ( x-frac{15}{4}=0), but this step does not get the variable alone on one side of the equation. The correct answer is: Add ( frac{1}{4}) to both sides of the equation.

  4. Add ( frac{7}{2}) to both sides of the equation.

    Incorrect. Adding ( frac{7}{2}) to both sides will result in the equivalent expression ( x-frac{1}{4}+frac{7}{2}=frac{7}{2}+frac{7}{2}), which can be rewritten ( x-frac{1}{4}+frac{14}{4}=frac{14}{2}), or ( x+frac{13}{4}=7), but this step does not get the variable alone on one side of the equation. The correct answer is: Add ( frac{1}{4}) to both sides of the equation.

With any equation, you can check your solution by substituting the value for the variable in the original equation. In other words, you evaluate the original equation using your solution. If you get a true statement, then your solution is correct.

Example

Solve ( x+10=-65). Check your solution.

Solution

( x+10=-65)Since 10 is being added to the variable, subtract 10 from both sides.
( egin{aligned}
x+10=&-65
-10 &-10 \
hline x=&-75
end{aligned})
Note that subtracting 10 is the same as adding -10. You get ( x=-75).
Check:( egin{aligned}
x+10 &=-65
-75+10 &=-65 \
-65 &=-65
end{aligned})

To check, substitute the solution, -75 for ( x) in the original equation.

Simplify. This equation is true, so the solution is correct.

( x=-75) is the solution to the equation

( x+10=-65).

It is always a good idea to check your answer whether it is requested or not.

Just as you can add or subtract the same exact quantity on both sides of an equation, you can also multiply both sides of an equation by the same quantity to write an equivalent equation. Let’s look at a numeric equation, ( 5 cdot 3=15), to start. If you multiply both sides of this equation by 2, you will still have a true equation.

( egin{aligned}
5 cdot 3 &=15
5 cdot 3 cdot 2 &=15 cdot 2
30 &=30
end{aligned})

This characteristic of equations is generalized in the multiplication property of equality.

Multiplication Property of Equality

For all real numbers ( a), ( b), and ( c): If ( a=b), then ( a cdot c=b cdot c) (or ( a b=a c)).

If two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.

When the equation involves multiplication or division, you can “undo” these operations by using the inverse operation to isolate the variable. When the operation is multiplication or division, your goal is to change the coefficient to 1, the multiplicative identity.

Example

Solve ( 3 x=24). Check your solution.

Solution

( 3 x=24)Divide both sides of the equation by 3 to isolate the variable (have a coefficient of 1).
( egin{array}{l}
frac{3 x}{3}&=frac{24}{3}
x&=8
end{array})
Dividing by 3 is the same as having multiplied by ( frac{1}{3}).
( egin{aligned}
3 x &=24
3 cdot 8 &=24
24 &=24
end{aligned})
Check by substituting your solution, 8, for the variable in the original equation.

The solution is correct!

( x=8)

You can also multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to 1!

Example

Solve ( frac{1}{2} x=8). Check your solution.

Solution

( egin{array}{r}
frac{1}{2} x=8
2left(frac{1}{2} x ight)=2(8)
end{array})

The coefficient of ( frac{1}{2} x) is ( frac{1}{2}).

Since the multiplicative inverse of ( frac{1}{2}) is 2, you can multiply both sides of the equation by 2 to get a coefficient of 1 for the variable.

( egin{aligned}
frac{2}{2} x &=16
x &=16
end{aligned})
Multiply.
( frac{1}{2}(16)=8)Check by substituting your solution into the original equation.
( egin{array}{c}
frac{16}{2}=8
8=8
end{array})
The solution is correct!

( x=16)

Example

Solve ( left(-frac{1}{4} ight) x=2). Check your solution.

Solution

( left(-frac{1}{4} ight) x=2)The coefficient of the variable is ( -frac{1}{4}). Multiply both sides by the multiplicative inverse of ( -frac{1}{4}), which is -4.
( (-4)left(-frac{1}{4} ight) x=(-4) 2)Multiply.
( egin{array}{r}
ext { (1) } x=-8
x=-8
end{array})
Any number multiplied by its multiplicative inverse is equal to 1, so ( x=-8).
( egin{array}{r}
left(-frac{1}{4} ight)(-8)=2
frac{8}{4}=2
2=2
end{array})

Check by substituting your solution into the original equation.

The solution is correct.

( x=-8)

Advanced Example

Solve ( -frac{7}{2}=frac{x}{10}). Check your solution.

Solution

( egin{aligned}
10left(-frac{7}{2} ight) &=10left(frac{x}{10} ight)
-frac{70}{2} &=frac{10 x}{10}
-frac{70}{2} &=x
-35 &=x
end{aligned})

This problem contains two fractions. Multiply both sides by 10 in order to isolate the variable ( x).

Then simplify the fractions.

Check( egin{aligned}
-frac{7}{2} &=frac{x}{10}
-frac{7}{2} &=frac{-35}{10}
-frac{7}{2} cdot frac{5}{5} &=frac{-35}{10}
-frac{35}{10} &=frac{-35}{10}
end{aligned})

Check your answer by substituting -35 in for ( x).

The solution is correct.

( x=-35)

Exercise

Solve for ( x): ( 5 x=-100)

  1. ( x=20)
  2. ( x=-20)
  3. ( x=500)
  4. ( x=-500)
Answer
  1. ( x=20)

    Incorrect. To isolate the variable, you can divide both sides by 5, because any number divided by itself is 1. ( frac{-100}{5}=-20). The correct answer is -20.

  2. ( x=-20)

    Correct. Dividing both sides by 5, you find that ( x=-20). To check this answer, you can substitute -20 in for ( x) in the original equation to get a true statement: ( 5(-20)=-100).

  3. ( x=500)

    Incorrect. You probably multiplied both sides by -5. To isolate the variable, remember to divide, not multiply, both sides by 5 because any number divided by itself is 1. The correct answer is -20.

  4. ( x=-500)

    Incorrect. You probably multiplied both sides by 5. To isolate the variable, remember to divide, not multiply, both sides by 5 because any number divided by itself is 1. The correct answer is -20.

Advanced Question

Solve for ( y): ( 4.2=7 y)

  1. ( y=0.6)
  2. ( y=29.4)
  3. ( y=1.67)
  4. ( y=-2.8)
Answer
  1. ( y=0.6)

    Correct. To isolate the variable ( y) on the right side of the equation, you have to divide both sides of the equation by ( ext { 7. } frac{4.2}{7}=0.6), so ( y=0.6).

  2. ( y=29.4)

    Incorrect. It looks like you multiplied ( 4.2 cdot 7) to get an answer of 29.4. Remember that if you multiply the left side by 7, you also have to multiply the right by 7; this does not isolate the variable! The correct answer is ( y=0.6).

  3. ( y=1.67)

    Incorrect. It looks like you divided 7 by 4.2 to get an answer of 1.67. However, dividing both sides by 4.2 does not isolate the variable! The correct answer is ( y=0.6).

  4. ( y=-2.8)

    Incorrect. It looks like you subtracted 7 from 4.2 to get an answer of -2.8. Remember that the expression ( 7y) means “7 times ( y).” To isolate the variable, you need to use the inverse of multiplication, not subtraction. The correct answer is ( y=0.6).

Equations are mathematical statements that combine two expressions of equal value. An algebraic equation can be solved by isolating the variable on one side of the equation using the properties of equality. To check the solution of an algebraic equation, substitute the value of the variable into the original equation.


Solve 1-Step Equations



Examples, solutions, videos, and lessons to help High School students learn how to explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Suggested Learning Targets

  • Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process. Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc.

Solving and explaining simple algebraic equations.
Recognize that properties of equality apply to equations containing unknown values by making connections to real numbers.
How to solve equations by using multiplicative and additive inverses?

Solving One Step Equations: The Basic
This video explains how to solve basic one step equations.
Examples:
Solve each equation.
1. x - 8 = 21
2. x + 4 = 11
3. 9 = x - 10
4. 17 = x + 2
5. x - 5 = -16
6. x + 4 = -17

Solve each equation.
1. x - 8 = 15
2. y + 23 = 41
3. 43 = x + 9
4. 23 = y - 15

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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One-Step Equations

Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the subtraction property of equality, and it lets you subtract the same number from both sides of an equation. Watch the video to see it in action!

What's the Addition Property of Equality?

Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the addition property of equality, and it lets you add the same number to both sides of an equation. Watch the video to see it in action!

What's the Multiplication Property of Equality?

Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the multiplication property of equality, and it lets you multiply both sides of an equation by the same number. Watch the video to see it in action!

How Do You Solve a Word Problem with an Equation Using Addition?

Word problems are a great way to see math in action! See how to translate a word problem into an equation, solve to find the answer, and check your found answer all in this tutorial.

How Do You Solve an Equation Using Addition?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve a Word Problem with an Equation Using Subtraction?

Word problems are a great way to see math in action! See how to translate a word problem into an equation, solve to find the answer, and check your found answer all in this tutorial.

How Do You Solve a Word Problem Using an Equation Where You're Multiplying Fractions?

Working with word problems AND fractions? This tutorial shows you how to take a word problem and translate it into a mathematical equation involving fractions. Then, you'll see how to solve and get the answer. Check it out!

How Do You Solve an Equation Where You're Multiplying Fractions?

Solving an equation with multiple fractions in different forms isn't so bad. This tutorial shows you how to convert a mixed fraction to an improper fraction in order to solve the equation. Then, you'll see how to convert the answer back to a mixed fraction to make sense of it. Follow along with this tutorial to see how it's done!

How Do You Solve a Word Problem with an Equation Using Multiplication?

Working with word problems AND fractions? This tutorial shows you how to take a word problem and translate it into a mathematical equation involving fractions. Then, you'll see how to solve and get the answer. Check it out!

How Do You Solve an Equation Using Multiplication?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve a Word Problem with an Equation Using Division?

Word problems are a great way to see math in action! See how to translate a word problem into an equation, solve to find the answer, and check your found answer all in this tutorial.

How Do You Solve an Equation by Guessing and Checking?

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How Do You Solve an Equation Using Division?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve an Equation Using Subtraction?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

What's the Division Property of Equality?

Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the division property of equality, and it lets you divide both sides of an equation by the same number. Watch the video to see it in action!

How Do You Solve an Equation With Fractions With The Same Denominators Using Subtraction?

Looking for practice solving equations containing fractions? Then check out this tutorial! Follow along and see how to subtract fractions with common denominators in order to solve an equation for a variable.

How Do You Solve an Equation With Fractions With The Same Denominators Using Addition?

Looking for practice solving equations containing fractions? Then check out this tutorial! Follow along and see how to add fractions with common denominators in order to solve an equation for a variable.

How Do You Solve an Equation With Fractions With Different Denominators Using Subtraction?

Looking for practice solving equations containing fractions? In this tutorial, you'll see how to first convert a mixed fraction to an improper fraction and then subtract fractions with unlike denominators in order to solve an equation. Be sure to check you answers so you KNOW it's correct!

How Do You Solve an Equation With Fractions With Different Denominators Using Addition?

Looking for practice solving equations containing fractions? In this tutorial, see how to add fractions with unlike denominators in order to solve an equation. Then, be sure to check you answers so you KNOW it's correct!

How Do You Solve a Decimal Equation Using Subtraction?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to subtract decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Addition?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to add decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Division?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to divide decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Multiplication?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to multiply decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!


Solving One-Step Equation Worksheets

One-step equation worksheets have exclusive pages to solve the equations involving fractions, integers, and decimals. Perform the basic arithmetic operations - addition, subtraction, multiplication and division to solve the equations. Exercises on the application of the equations in real life are available here to impart practical knowledge. This set of printable worksheets is specially designed for 6th grade, 7th grade, and 8th grade students. Free sample worksheets are included.

A variety of one-step equations involving all the four basic operations are given in these mixed operation pdf worksheets. Perform the appropriate operation and solve for the unknown variable.

Taking your practice a step higher, the coefficients are rendered in positive and negative integers. Retain the variable on one side, take the coefficient and constant to the other side and solve.

Add, subtract, multiply, and divide to solve the one-step fraction equations in these level 1 worksheets that involve proper and improper fractions as coefficients and constants.

A moderate practice awaits 7th grade and 8th grade students here! Solve a series of one-step equations with their terms incorporating fractions as well as mixed numbers.

The terms of the one-step equations in these worksheets are either decimals or integers. All four arithmetic operations are involved here to solve the problems.

In these printable worksheets, the coefficient of each one-step equation may be an integer, fraction or decimal. Complete practice can be given to children by solving these equations.

Employ this assembly of one-step equation word problems featuring integers, decimals and fraction coefficients.

Solve each one-step equation to find the unknown variable. Substitute the value of the variable in the given equation to verify the solution of the equation. This set of worksheets is ideal for students of grade 7 and grade 8.

Plenty of multiple choice questions are available in these handouts. Solve the indicated equations and choose the correct integer values from the given options.

Solving equations, finding the equation with a given solution, and evaluating expressions with the obtained values are the skills you can acquire in these pdf MCQ worksheets featuring fractions.

These printable worksheets contain an activity based exercise to find the cost of the products. The price tags of the objects are represented in an equation form. Solve the equations.

Children in grade 6 should read each verbal phrases / sentences and translate it to an appropriate one-step linear equation.

Guess my number! These fun math riddles help kids to easily understand and translate the sentences into equations. Try all these interesting problems.

Enhance your knowledge by solving these one-step equations on geometry. In 'Type 1' pdf worksheets, find the unknown sides of the given shape by solving the one-step equations.

'Type 2' printable worksheets contain problems based on applications in geometry. Apply the properties of shapes to find the unknown parameter(s).


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Two-Step Equations and Properties of Equality

To check solutions to two step equations, we put our solution back into the equation and check that both sides equal.

If they equal, then we know our solution is correct. If not, then our solution is wrong.

First step: Multiply both sides by 2:

Second step: Subtract #2x# from both sides:

Solving two-step equations is not much more complicated than solving one-step equations it just involves an extra step.

Usually there is more than one way to solve these. It's ok to use whatever method makes most sense to you. The general rule of thumb when isolating the variable is to undo the order of operations, PEMDAS. Start with addition and subtraction, then multiplication and division, then exponents, and finally parentheses.

Let's look at an example: # 2x - 6 = 12#

#2x-6=12 #
#2x -6 +6 = 12 + 6 # add 6 to each side
#2x = 18 #
#(2x)/2=18/2# divide each side by 2
#x=9 #

#2x -6 = 12 #
#(2x-6)/2= 12/2 # divide each side by 2
#(2x)/2 - 6/2 =12/2# separate the fractions
#x - 3 = 6# simplify
#x - 3 + 3 = 6 + 3 # add 3 to each side
#x=9 #

Step 1) Add or Subtract the necessary term from each side of the equation to isolate the term with the variable while keeping the equation balanced.

Step 2) Mulitply or Divide each side of the equation by the appropriate value solve for the variable while keeping the equation balanced.


Solving One-Step Equations

Word problems are a great way to see math in action! See how to translate a word problem into an equation, solve to find the answer, and check your found answer all in this tutorial.

How Do You Solve an Equation Using Addition?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve a Word Problem with an Equation Using Subtraction?

Word problems are a great way to see math in action! See how to translate a word problem into an equation, solve to find the answer, and check your found answer all in this tutorial.

How Do You Solve a Word Problem Using an Equation Where You're Multiplying Fractions?

Working with word problems AND fractions? This tutorial shows you how to take a word problem and translate it into a mathematical equation involving fractions. Then, you'll see how to solve and get the answer. Check it out!

How Do You Solve an Equation Where You're Multiplying Fractions?

Solving an equation with multiple fractions in different forms isn't so bad. This tutorial shows you how to convert a mixed fraction to an improper fraction in order to solve the equation. Then, you'll see how to convert the answer back to a mixed fraction to make sense of it. Follow along with this tutorial to see how it's done!

How Do You Solve a Word Problem with an Equation Using Multiplication?

Working with word problems AND fractions? This tutorial shows you how to take a word problem and translate it into a mathematical equation involving fractions. Then, you'll see how to solve and get the answer. Check it out!

How Do You Solve an Equation Using Multiplication?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve a Word Problem with an Equation Using Division?

Word problems are a great way to see math in action! See how to translate a word problem into an equation, solve to find the answer, and check your found answer all in this tutorial.

How Do You Solve an Equation by Guessing and Checking?

Want to solve an equation by guessing and checking possible answers? Then this tutorial is for you! Make sure to pay close attention to the strategy involved in guessing and checking!

How Do You Solve an Equation Using Division?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve an Equation Using Subtraction?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve an Equation With Fractions With The Same Denominators Using Subtraction?

Looking for practice solving equations containing fractions? Then check out this tutorial! Follow along and see how to subtract fractions with common denominators in order to solve an equation for a variable.

How Do You Solve an Equation With Fractions With The Same Denominators Using Addition?

Looking for practice solving equations containing fractions? Then check out this tutorial! Follow along and see how to add fractions with common denominators in order to solve an equation for a variable.

How Do You Solve an Equation With Fractions With Different Denominators Using Subtraction?

Looking for practice solving equations containing fractions? In this tutorial, you'll see how to first convert a mixed fraction to an improper fraction and then subtract fractions with unlike denominators in order to solve an equation. Be sure to check you answers so you KNOW it's correct!

How Do You Solve an Equation With Fractions With Different Denominators Using Addition?

Looking for practice solving equations containing fractions? In this tutorial, see how to add fractions with unlike denominators in order to solve an equation. Then, be sure to check you answers so you KNOW it's correct!

How Do You Solve a Decimal Equation Using Subtraction?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to subtract decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Addition?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to add decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Division?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to divide decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Multiplication?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to multiply decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!


Two-Step Equations and Properties of Equality

To check solutions to two step equations, we put our solution back into the equation and check that both sides equal.

If they equal, then we know our solution is correct. If not, then our solution is wrong.

First step: Multiply both sides by 2:

Second step: Subtract #2x# from both sides:

Solving two-step equations is not much more complicated than solving one-step equations it just involves an extra step.

Usually there is more than one way to solve these. It's ok to use whatever method makes most sense to you. The general rule of thumb when isolating the variable is to undo the order of operations, PEMDAS. Start with addition and subtraction, then multiplication and division, then exponents, and finally parentheses.

Let's look at an example: # 2x - 6 = 12#

#2x-6=12 #
#2x -6 +6 = 12 + 6 # add 6 to each side
#2x = 18 #
#(2x)/2=18/2# divide each side by 2
#x=9 #

#2x -6 = 12 #
#(2x-6)/2= 12/2 # divide each side by 2
#(2x)/2 - 6/2 =12/2# separate the fractions
#x - 3 = 6# simplify
#x - 3 + 3 = 6 + 3 # add 3 to each side
#x=9 #

Step 1) Add or Subtract the necessary term from each side of the equation to isolate the term with the variable while keeping the equation balanced.

Step 2) Mulitply or Divide each side of the equation by the appropriate value solve for the variable while keeping the equation balanced.


The Issues and Solutions

1. Not Showing Work

As good math teachers, start by exposing the students to simpler problems to teach the technique, and then we increase the rigor.

The issue this causes is that most of your students are smart enough to figure out the solution without doing the algebra needed to solve. Thus, when the problems get harder, the students do not have the tools to be able to solve the problem – then you must ‘reteach’ them the skill (though, it’s not really a reteach, since they never actually learned it.)

One easy way to combat this issue it to make your students show the work on the easier problems. Explain to them that you know that they know the answer without the work – but the purpose of this activity is not to see if they know what minus 1 equals 8, the purpose of this activity is to learn how to solve a one-step equation. I often remind my students as they are learning the skill that the problems will get harder, so learn the steps now or you won’t be able to do the work later.

Another easy method to help deter students from skipping the process of showing their work (or not learning how to properly do the steps) is to expose them to the harder questions at the beginning. This way your students understand why they need to learn how to show their work.

2. Lack of prerequisite skills

If your students did not master adding integers, subtracting integers, or multiplying and dividing integers, they will not be able to solve one-step equations.

Often, one-step equations have fractions or decimals involved, so your students must also know how to perform these skills as well.

Since these prerequisite skills will hinder my students from mastering this concept, I remediate my students based on what skill they have not mastered ”skills based remediation.”

Since different students will need remediation on different skills, I build a learning station for each skill that reteaches the math concept, and then exposes the students to lots of practice with immediate feedback. We do station work every day, and my students stay in the station of the skill that they need remediation on until they master that skill. Then I progress them to the next, and then the next, until they can solve one-step equations.

Premade Learning Stations

3. Not understanding the equal sign

This may be a shocker for many teachers. It was for me when I first started witnessing it with my students. But many of our students don’t understand what the equal sign means.

The equal sign shows that the left side of the equation and the right side of the equation have the same value they’re balanced. This is why whatever we do to one side of the equation, we must do to the other side – to keep them balanced.

Because students don’t understand this, they fail to perform the same operation to both sides of the equal sign.

You may have seen students perform the same operation to one side of the equal sign, twice! Or they will get confused when the variable is to the right of the equal sign (because they think that the equal sign means “solve”).

How many times have you showed students how to check their work on a one-step equation and they don’t understand why you’re doing it. I was always surprised when I’d do the ‘check your work’ on the board, show that both sides of the same resulted in the same answer, “ 8 = 8 “ but when I asked my students if ‘eight equals eight’ they didn’t know. This was because they didn’t understand what an equal sign is.

Students think the equal sign means “solve.” So when I ask if “ 8=8?” They read it, ‘8 solve 8,’ which of course makes no sense.

The first thing you should do to combat this issue is teach your students what the equal sign means. Teach it every day, several times a day. Ask them as they’re doing their work, entering the room, or when they ask to go to the bathroom, “What does the equal sign mean?” Engrain it into their thinking, “the equal sign means both sides of the equation have the same value.” Obviously you can use a different vocabulary to convey the same message,

The second thing you must do is make sure that you are exposing your students to problems that are not in the traditional equation format.

The traditional formatting of an equation is: 7 + x = 9

So rearrange it: x + 7 = 9 and 9 = x + 7

To go one step further, you can even show the traditional formatting of the equation to your students, and then ask them to tell it to you in a different setup.

The final method I recommend to combat this misunderstanding of the equal sign is a math game that helps students understand this concept and think about it critically. To play, you need two dice for every group of students and a piece of paper.

The game is a lot of fun, but it also helps students better understand how the equal sign works.

Want to learn more about using learning stations in your class? Click here

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One-Step Equations

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What's the Addition Property of Equality?

Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the addition property of equality, and it lets you add the same number to both sides of an equation. Watch the video to see it in action!

What's the Multiplication Property of Equality?

Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the multiplication property of equality, and it lets you multiply both sides of an equation by the same number. Watch the video to see it in action!

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How Do You Solve a Word Problem with an Equation Using Subtraction?

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How Do You Solve a Word Problem Using an Equation Where You're Multiplying Fractions?

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How Do You Solve an Equation Where You're Multiplying Fractions?

Solving an equation with multiple fractions in different forms isn't so bad. This tutorial shows you how to convert a mixed fraction to an improper fraction in order to solve the equation. Then, you'll see how to convert the answer back to a mixed fraction to make sense of it. Follow along with this tutorial to see how it's done!

How Do You Solve a Word Problem with an Equation Using Multiplication?

Working with word problems AND fractions? This tutorial shows you how to take a word problem and translate it into a mathematical equation involving fractions. Then, you'll see how to solve and get the answer. Check it out!

How Do You Solve an Equation Using Multiplication?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve a Word Problem with an Equation Using Division?

Word problems are a great way to see math in action! See how to translate a word problem into an equation, solve to find the answer, and check your found answer all in this tutorial.

How Do You Solve an Equation by Guessing and Checking?

Want to solve an equation by guessing and checking possible answers? Then this tutorial is for you! Make sure to pay close attention to the strategy involved in guessing and checking!

How Do You Solve an Equation Using Division?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

How Do You Solve an Equation Using Subtraction?

Solving an equation for a variable? Perform the order of operations in reverse! Check it out in this tutorial.

What's the Division Property of Equality?

Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the division property of equality, and it lets you divide both sides of an equation by the same number. Watch the video to see it in action!

How Do You Solve an Equation With Fractions With The Same Denominators Using Subtraction?

Looking for practice solving equations containing fractions? Then check out this tutorial! Follow along and see how to subtract fractions with common denominators in order to solve an equation for a variable.

How Do You Solve an Equation With Fractions With The Same Denominators Using Addition?

Looking for practice solving equations containing fractions? Then check out this tutorial! Follow along and see how to add fractions with common denominators in order to solve an equation for a variable.

How Do You Solve an Equation With Fractions With Different Denominators Using Subtraction?

Looking for practice solving equations containing fractions? In this tutorial, you'll see how to first convert a mixed fraction to an improper fraction and then subtract fractions with unlike denominators in order to solve an equation. Be sure to check you answers so you KNOW it's correct!

How Do You Solve an Equation With Fractions With Different Denominators Using Addition?

Looking for practice solving equations containing fractions? In this tutorial, see how to add fractions with unlike denominators in order to solve an equation. Then, be sure to check you answers so you KNOW it's correct!

How Do You Solve a Decimal Equation Using Subtraction?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to subtract decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Addition?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to add decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Division?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to divide decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!

How Do You Solve a Decimal Equation Using Multiplication?

Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to multiply decimals in order to solve an equation for a variable. Then, see how to check your answer so you can be certain it's correct!


Watch the video: Der Satz vom Nullprodukt Gleichungen umstellen. lösen Gehe auf (October 2021).