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10.1: Solving Equations - Mathematics


10.1: Solving Equations - Mathematics

How to Solve Equations with Fractions?

Do not get panic when you see fractions in an algebraic equation. If you know all the rules for adding, subtracting, multiplying, and dividing, it’s a piece of cake for you.

To solve equations with fractions, you need to transform them into an equation without fractions.

This method is also called the “clearing of fractions.”

In solving equations with fractions, the following steps are followed:

  • Determine the lowest common multiple of the denominators (LCD) of all fractions in an equation and multiply by all fractions in the equation.
  • Isolate the variable.
  • Simplify both sides of an equation by applying simple algebraic operations.
  • Apply division or multiplication property to make the coefficient of a variable equal to 1.

The LCD of 5 and 3 is 15, therefore multiply both
(3x + 4)/5 = (2x – 3)/3

The LCD of 2x, 4, and 3 is 12x

Multiply each fraction in the equation by the LCD.

Solve for x (2 + 2x)/4 = (1 + 2x)/8

Multiply each fraction by the LCD,

Practice Questions

1. Solve for x in the following linear equations:

2. Jared’s age is four times as old as his son. After 5 years, Jared will be 3 times as old as his son. Find the present age of Jared and his son.

3. The cost of 2 pairs of trousers and 3 shirts is $705. If a shirt costs $40 less than a pair of trousers, find each shirt and trouser’s cost.

4. A boat takes 6 hours when sailing upstream and 5 hours when sailing downstream of a river. Calculate the boat’s speed in still water given that the speed of the river is 3 km/hour.

5. A two-digit number has the sum of its digits is 7. When the digits are reversed, the number formed is 27 less than the original number. Find the number.

6. $ 10000 is distributed among 150 people. If the money is either in the denomination of $100 or $50. Calculate the number of each denomination of the money.

7. The width of a rectangle is 3cm less than the length. When the width and length are enlarged by 2, the rectangle area changes to 70 cm 2 more than that of the original rectangle. Calculate the dimensions of the original rectangle.

8. The numerator of a fraction 8 less than the denominator. When the denominator is reduced by 1 and the numerator increased by 17, the fraction becomes 3/2. Determine the fraction.

9. My father is 12 years more than two times my age. After 8 years, my father’s age will be 20 less than 3 times my age. What is my father’s present age?


10.1 Solve Quadratic Equations Using the Square Root Property

We have seen that some quadratic equations can be solved by factoring. In this chapter, we will use three other methods to solve quadratic equations.

Solve Quadratic Equations of the Form ax 2 = k Using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation x 2 = 9 x 2 = 9 .

This leads to the Square Root Property .

Square Root Property

Notice that the Square Root Property gives two solutions to an equation of the form x 2 = k x 2 = k : the principal square root of k k and its opposite. We could also write the solution as x = ± k x = ± k .

What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation x 2 = 7 x 2 = 7 .

Example 10.1

Solution

Use the Square Root Property. Simplify the radical. x 2 = 169 x = ± 169 x = ± 13 Rewrite to show two solutions. x = 13 , x = −13 Use the Square Root Property. Simplify the radical. x 2 = 169 x = ± 169 x = ± 13 Rewrite to show two solutions. x = 13 , x = −13

Example 10.2

How to Solve a Quadratic Equation of the Form a x 2 = k a x 2 = k Using the Square Root Property

Solution

How To

Solve a quadratic equation using the Square Root Property.

  1. Step 1. Isolate the quadratic term and make its coefficient one.
  2. Step 2. Use Square Root Property.
  3. Step 3. Simplify the radical.
  4. Step 4. Check the solutions.

To use the Square Root Property, the coefficient of the variable term must equal 1. In the next example, we must divide both sides of the equation by 5 before using the Square Root Property.

Example 10.3

Solution

Example 10.4

Solution

Remember, we first isolate the quadratic term and then make the coefficient equal to one.

Example 10.5

Solution

The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator.

Example 10.6

Solution




Isolate the quadratic term.

Divide by 2 to make the coefficient 1.



Simplify.

Use the Square Root Property.

Simplify the radical.

Rationalize the denominator.

Simplify.
2 c 2 − 4 = 45 2 c 2 = 49 2 c 2 2 = 49 2 c 2 = 49 2 c = ± 49 2 c = ± 49 2 c = ± 49 · 2 2 · 2 c = ± 7 2 2 2 c 2 − 4 = 45 2 c 2 = 49 2 c 2 2 = 49 2 c 2 = 49 2 c = ± 49 2 c = ± 49 2 c = ± 49 · 2 2 · 2 c = ± 7 2 2
Rewrite to show two solutions. c = 7 2 2 , c = − 7 2 2 c = 7 2 2 , c = − 7 2 2
Check. We leave the check for you.

Solve Quadratic Equations of the Form a(x − h) 2 = k Using the Square Root Property

We can use the Square Root Property to solve an equation like ( x − 3 ) 2 = 16 ( x − 3 ) 2 = 16 , too. We will treat the whole binomial, ( x − 3 ) ( x − 3 ) , as the quadratic term.

Example 10.7

Solution

Example 10.8

Solution

Remember, when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.

Example 10.9

Solution





Use the Square Root Property.

Rewrite the radical as a fraction of square roots.


Simplify the radical.


Solve for x.
( x − 1 2 ) 2 = 5 4 x − 1 2 = ± 5 4 x − 1 2 = ± 5 4 x − 1 2 = ± 5 2 x = 1 2 ± 5 2 ( x − 1 2 ) 2 = 5 4 x − 1 2 = ± 5 4 x − 1 2 = ± 5 4 x − 1 2 = ± 5 2 x = 1 2 ± 5 2
Rewrite to show two solutions. x = 1 2 + 5 2 , x = 1 2 − 5 2 x = 1 2 + 5 2 , x = 1 2 − 5 2
Check. We leave the check for you.

We will start the solution to the next example by isolating the binomial.

Example 10.10

Solution

Example 10.11

Solution

The left sides of the equations in the next two examples do not seem to be of the form a ( x − h ) 2 a ( x − h ) 2 . But they are perfect square trinomials, so we will factor to put them in the form we need.

Example 10.12

Solution

The left side of the equation is a perfect square trinomial. We will factor it first.




Factor the perfect square trinomial.

Use the Square Root Property.

Simplify the radical.

Solve for p.
p 2 − 10 p + 25 = 18 ( p − 5 ) 2 = 18 p − 5 = ± 18 p − 5 = ± 3 2 p = 5 ± 3 2 p 2 − 10 p + 25 = 18 ( p − 5 ) 2 = 18 p − 5 = ± 18 p − 5 = ± 3 2 p = 5 ± 3 2
Rewrite to show two solutions. p = 5 + 3 2 , p = 5 − 3 2 p = 5 + 3 2 , p = 5 − 3 2
Check. We leave the check for you.

Example 10.13

Solution

Again, we notice the left side of the equation is a perfect square trinomial. We will factor it first.

Solve: 16 n 2 + 40 n + 25 = 4 16 n 2 + 40 n + 25 = 4 .

Media

Access these online resources for additional instruction and practice with solving quadratic equations:

Section 10.1 Exercises

Practice Makes Perfect

Solve Quadratic Equations of the form a x 2 = k a x 2 = k Using the Square Root Property

In the following exercises, solve the following quadratic equations.

Solve Quadratic Equations of the Form a ( x − h ) 2 = k a ( x − h ) 2 = k Using the Square Root Property

In the following exercises, solve the following quadratic equations.

Mixed Practice

In the following exercises, solve using the Square Root Property.

Everyday Math

Paola has enough mulch to cover 48 square feet. She wants to use it to make three square vegetable gardens of equal sizes. Solve the equation 3 s 2 = 48 3 s 2 = 48 to find s s , the length of each garden side.

Kathy is drawing up the blueprints for a house she is designing. She wants to have four square windows of equal size in the living room, with a total area of 64 square feet. Solve the equation 4 s 2 = 64 4 s 2 = 64 to find s s , the length of the sides of the windows.

Writing Exercises

Explain why the equation x 2 + 12 = 8 x 2 + 12 = 8 has no solution.

Explain why the equation y 2 + 8 = 12 y 2 + 8 = 12 has two solutions.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently: Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help: This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no-I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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    Chapter 10.1: DeepNLP — LSTM (Long Short Term Memory) Networks with Math.

    Note: I am writing this article with the assumption that you know the deep learning a bit. In case if you don’t know much, Please read my earlier stories to understand the entire series on deep learning.

    In the last story we talked about Recurrent neural networks, so we now know what RNN’s are, How they work and what kind of problems it can solve and also we talked about a limitation in RNN’s which is

    Vanishing /exploding gradient problem

    We all know that a neural network uses an algorithm called BackPropagation to update the weights of the network. So what BP does is

    It first calculates the gradients from the error using the chain rule in Calculus, then it updates the weights(Gradient descent).

    since the BP starts from the output layer to all the way back to input layer , In a simple neural network we may not face problems with updating weights but in a deep neural network we might face some issues.

    As we go back with the gradients , It is possible that the values get either smaller exponentially which causes Vanishing Gradient problem or larger exponentially which causes Exploding Gradient problem.

    Due to this we get the problems of training the network.

    In RNN’s, we have time steps and current time step value depends on the previous time step so we need to go all the way back to make an update.

    There are couple of remedies there to avoid this problem.

    We can use ReLu unit as an activation function, RMS Prop as an optimization algorithm and LSTM’s or GRU’s.

    LSTM ( Long Short Term Memory ) Networks are called fancy recurrent neural networks with some additional features.

    Just like RNN, we have time steps in LSTM but we have extra piece of information which is called “MEMORY” in LSTM cell for every time step.

    So the LSTM cell contains the following components

    1. Forget Gate “f” ( a neural network with sigmoid)
    2. Candidate layer “C`"(a NN with Tanh)
    3. Input Gate “I” ( a NN with sigmoid )
    4. Output Gate “O”( a NN with sigmoid)
    5. Hidden state “H” ( a vector )
    6. Memory state “C” ( a vector)

    Here is the diagram for LSTM cell at the time step t

    Don’t panic I will explain every single hecking detail of it. Just get the overall picture stored in your brain.

    Lemme take only one time step (t) and explain it.

    What are the inputs and outputs of the LSTM cell at any step ??

    Inputs to the LSTM cell at any step are X (current input) , H ( previous hidden state ) and C ( previous memory state)

    Outputs from the LSTM cell are H ( current hidden state ) and C ( current memory state)

    Here is the diagram for a LSTM cell at T time step.

    If you observe carefully,the above diagram explains it all.

    Anyway, lemme also try with words

    Forget gate(f) , Cndate(C`), Input gate(I), Output Gate(O)

    are single layered neural networks with the Sigmoid activation function except candidate layer( it takes Tanh as activation function)

    These gates first take input vector.dot(U) and previous hidden state.dot(W) then concatenate them and apply activation function

    finally these gate produce vectors ( between 0 and 1 for Sigmoid, -1 to 1 for Tanh) so we get four vectors f, C`, I, O for every time step.

    Now let me tell you an important piece called Memory state C

    This is the state where the memory (context) of input is stored

    Ex : Mady walks in to the room, Monica also walks in to the room. Mady Said “hi” to ____??

    Inorder to predict correctly Here it stores “Monica” into memory C.

    This state can be modified. I mean LSTM cell can add /remove the information.

    Ex : Mady and Monica walk in to the room together , later Richard walks in to the room. Mady Said “hi” to ____??

    The assumption I am making is memory might change from Monica to Richard.

    so LSTM cell takes the previous memory state Ct-1 and does element wise multiplication with forget gate (f)

    if forget gate value is 0 then previous memory state is completely forgotten

    if forget gate value is 1 then previous memory state is completely passed to the cell ( Remember f gate gives values between 0 and 1 )

    Now with current memory state Ct we calculate new memory state from input state and C layer.

    Ct= Ct + (It*C`t)

    Ct = Current memory state at time step t. and it gets passed to next time step.

    Here is flow diagram for Ct

    Finally, we need to calculate what we’re going to output. This output will be based on our cell state Ct but will be a filtered version. so we apply Tanh to Ct then we do element wise multiplication with the output gate O, That will be our current hidden state Ht

    We pass these two Ct and Ht to the next time step and repeat the same process.


    47 thoughts on &ldquo 10 Mind Blowing Mathematical Equations &rdquo

    Brilliant dude! I had a great time looking a lot of these up that I didn’t know about before. One thing that you didn’t mention is the sheer simplicity of many of these, a factor which really contributes to their beauty. I had no idea that e and pi could be related as simply as the Gaussian integral illustrates for example. (It also brings into question the fact that they are considered two separate fundamental constants of the universe.) And the Euler Product Formula is also great for its simplicity

    are u mad bro? No one likes maths…I bet you were one of those people that got their heads flushed down the lavoratory…good day to you!

    If you don’t like math, why are you here? That is exceedingly redundant you incompetent moron.

    so much pretense. someone needs a brake from the thesaurus.

    I love math and i am only 12

    Are U mad? Lots of people like math… I bet U were one of those people that got their heads flushed down the lavoratory…good day to U!

    AGREED… I LOVE MATH… ITS FUN

    Y u mean just because someone else likes math doesn’t mean u can disown them. I bet that ur not even good at math. BTW if u didn’t like math y u click on this website

    Here’s a neat little ditty a friend of mine came up with.

    Yep, that is a good one! It is normally written as i^(-i) = sqrt(e^pi). It is remarkable that raising i to the -i power results a real number, let alone being related to e and pi.

    i think that would look nicer as e^pi=1/i^2i

    that’s wrong! the correct one is -3^5(55-d.o*

    This is just a different expression of Euler’s identity.

    What about the humble quadratic formula? Surely it deserves a mention.

    Well, the quadratic formula is quite beautiful, but isn’t as mind blowing as the ones listed. The quadratic formula is quite… expected and reasonable, if you know what I mean

    No, no, no, no, no! The quadratic or trinomial equation is NOT beautiful. The equation which proves that infinity=-1/12 is gorgeous!

    ITS FREAKING MIND BOGGLING.THANKS MAN!

    WELL CAN I GET RAMANUJAN’S INFNITE SERIES

    I don’t think i’ll ever come to terms with the Basel Problem

    Yeah, I’ve gone through two proofs of it (well, one was only a semi-proof), but the equality is not any more apparent.

    The proof of it doesn’t really bother me, that actually seems pretty clear, albeit gorgeous and innovative. The final result really just blows my mind though.

    I want reblog your post, bro. I like everything about Math, Specially about the beauty.

    If you think think you guys are so smart then what is 1+2=?

    I would say duh. but I knowq such questions do not ned the real answer.

    managing ur lyf x De best way 2 encourage othrs whlst u r in ur middle ages kk ..De Eula formula x De most formula ever I hv ever known…

    Hey math chaps this all seems funny i mean some high school maths ..yeah

    Re: Basel, it’s also cool (arguably slightly cooler?) to look at

    which converges very slowly, but steadily…. to exactly pi.

    This all equations are mindblowing. However, I think root of unity (Z^n=1) should have been accomodated in this list.

    To me Eulers identity does it (e(^pi*i)=0,which I believe it cannot be proved without using De moivre’s formula,that is ofcouse another mindblowiong formula.

    This one is neat: 1-1/2+1/4-1/8+1/16-1/32+1/64-1/128…….=1/3

    Great summary of the most beautiful identities !! Great job !

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    HOLY SHIT YOUR ALL FUCKING NERDS

    I come here searching for 10 Mind Blowing Mathematical Equations .
    Now, Mathematics comes from many different sorts of problems.
    Initially these were within commerce, land way of measuring, structures
    and later astronomy today, all sciences suggest problems examined by mathematicians, and many
    problems occur within mathematics itself. For instance, the physicist Richard Feynman created the path important formulation of quantum technicians utilizing a combo of mathematical reasoning and physical information, and today’s string theory, a
    still-developing technological theory which tries to unify the four important
    forces of aspect, continues to encourage new mathematics.

    Many mathematical items, such as packages of volumes and functions, display internal structure because of procedures or relationships that are described
    on the set in place. Mathematics then studies properties of these sets
    that may be expressed in conditions of that composition for instance quantity theory
    studies properties of the group of integers that may be expressed in conditions of arithmetic functions.
    Additionally, it frequently happens that different such organised sets (or constructions) show similar
    properties, rendering it possible, by an additional step of abstraction, to convey
    axioms for a school of buildings, and then research at once the complete class of set ups gratifying these axioms.

    Thus you can study groupings, rings, domains and other abstract systems mutually such studies
    (for set ups identified by algebraic procedures) constitute the domains of abstract algebra.

    Here: http://math-problem-solver.com To be able to clarify the foundations of mathematics,
    the areas of mathematical logic and place theory were developed.
    Mathematical logic includes the mathematical analysis of logic and the applications of formal logic to the areas of mathematics placed theory is the branch of mathematics that studies collections or series of items.

    Category theory, which offers within an abstract way with mathematical buildings
    and human relationships between them, continues to be
    in development.

    9 锟?5m fee with Monaco for midfield star Tiemoue Bakayoko on lucrative five-year dealinter and outInter Milan may be worst team ever for selling players before they reached their prime Paul Pogba posts funny instagram message for Antoine Griezmann's birthdayWayne Rooney 鈥?Manchester United to Everton 鈥?3.


    Equation and situations

    To rent a room in a certain building, you have to pay a deposit of R400 and then R80 per day.

    How much money do you need to rent the room for 10 days?

    How much money do you need to rent the room for 15 days?

    Which of the following best describes the method that you used to do question 1(a) and (b)?

    B. Total cost = 400(number of days + 80)

    C. Total cost = 80 ( imes) number of days + 400

    D. Total cost = (80 + 400) ( imes) number of days

    For how many days can you rent the room described in question 1, if you have R2 800 to pay?

    If you want to know for how many days you can rent the room if you have R720, you can set up an equation and solve it:

    You know the total cost is R720 and you know that you can work out the total cost like this:

    ( ext = 80x + 400), where (x) is the number of days. So, (80x + 400 = 720) and (x = 4) days.

    In each of the following cases, find the unknown number by setting up an equation and solving it.

    To rent a certain room, you have to pay a deposit of R300 and then R120 per day .

    For how many days can you rent the room if you can pay a total of R1 740? (If you experience trouble in setting up the equation, it may help you to decide first how you will work out what it will cost to rent the room for 6 days.)

    What will it cost to rent the room for 10 days, 11 days and 12 days?

    For how many days can you rent the room if you have R3 300 available?

    For how many days can you rent the room if you have R3 000 available?

    Ben and Thabo decide to do some calculations with a certain number. Ben multiplies the number by 5 and adds 12. Thabo gets the same answer as Ben when he multiplies the number by 9 and subtracts 16. What is the number they worked with?

    The cost of renting a certain car for a period of (x) days can be calculated with the following formula:

    What information about renting this car will you get, if you solve the equation

    Sarah paid a deposit of R320 for a stall at a market, and she also pays R70 per day rental for the stall. She sells fruit and vegetables at the stall, and finds that she makes about R150 profit each day. After how many days will she have earned as much as she has paid for the stall, in total?


    Solution

    When you solve this equation with pictures, you end up with 3 bags balancing with 1 tile. In order to do the division, you have to cut the tile, leading to the fraction 1/3, which is the solution you get symbolically.

    In order to solve this equation with pictures, you have to have some way of representing the subtraction in $2x – 4$. If students have experience with integer chips, they can transfer that knowledge to this situation to show $2x + -4$, but otherwise they may struggle with the idea. The pictures give us a nice model for understanding the operations we do to solve equations, but it is only smooth for problems with “nice” numbers. This is one reason why we want to move to the symbolic approach.

    A linear equation will have no solution if there are the same number of $x$’s and different constants on each side. For example: $2x + 4 = 2x + 1$. If you solve this with pictures, when you take away the $2x$ from both sides you will end up with $4 = 1$, which clearly cannot be balanced. If the equation had infinitely many solutions, you would find that you had exactly the same picture on the two sides of the balance.

    The mistake is in the first step - the student divided only part of the left-hand-side of the equation by 2. You can see in the picture that splitting the equation this way will not keep the balance level (assuming the two bags are equal):


    Equation Games

    On this page you can find a variety of fun math equation games that middle school and high school students can play online.

    Let's start with solving one-step equations.

    One-Step Equations with Addition and Subtraction
    This is a fun and interactive soccer math game about solving linear equations with whole numbers. All solutions are positive numbers.

    Math Basketball - One-Step Equations with Addition and Subtraction
    Play this interesting math basketball game and get points for scoring baskets and solving equations correctly.

    Solving One-Step Equations
    Did you know that solving equation can be exciting? Play these two games to find out how much fun you can have when solving one-step equations.

    Two-Step Equation Game
    Can you solve two-step equations with integers? Play this fun game to show off you skills.

    Equation Puzzle(New)
    This is an interactive crossword puzzle with key vocabulary words related to equations.

    Equation Word Search
    This is an interactive word search game that you can play online. It involves vocabulary words that people use when solving equations.

    Systems of Equations Game If you want to solve systems of equations and score tones of points, we have the perfect game for you. Click on the above link to check it out.

    Interactive Equation Game
    In this game, students must match different equations with their solutions as fast as possible.


    Watch the video: Solving systems of linear equations in two variables (October 2021).