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## Chapter 1: Solving Equations and Inequalities

Problem solving, learning objectives.

- Translate words into algebraic expressions and equations
- Define a process for solving word problems
- Apply the steps for solving word problems to distance, rate, and time problems
- Apply the steps for solving word problems to interest rate problems
- Evaluate a formula using substitution
- Rearrange formulas to isolate specific variables
- Identify an unknown given a formula
- Apply the steps for solving word problems to geometry problems
- Use the formula for converting between Fahrenheit and Celsius

## Define a Process for Problem Solving

- [latex]x\text{ is }5[/latex] becomes [latex]x=5[/latex]
- Three more than a number becomes [latex]x+3[/latex]
- Four less than a number becomes [latex]x-4[/latex]
- Double the cost becomes [latex]2\cdot\text{ cost }[/latex]
- Groceries and gas together for the week cost $250 means [latex]\text{ groceries }+\text{ gas }=250[/latex]
- The difference of 9 and a number becomes [latex]9-x[/latex]. Notice how 9 is first in the sentence and the expression

Let’s practice translating a few more English phrases into algebraic expressions.

Translate the table into algebraic expressions:

In this example video, we show how to translate more words into mathematical expressions.

- Read and understand the problem.
- Determine the constants and variables in the problem.
- Translate words into algebraic expressions and equations.
- Write an equation to represent the problem.
- Solve the equation.
- Check and interpret your answer. Sometimes writing a sentence helps.

Twenty-eight less than five times a certain number is 232. What is the number?

- Read and understand: we are looking for a number.
- Constants and variables: 28 and 232 are constants, “a certain number” is our variable because we don’t know its value, and we are asked to find it. We will call it x.
- Translate: five times a certain number translates to [latex]5x[/latex] Twenty-eight less than five times a certain number translates to [latex]5x-28[/latex] because subtraction is built backward. is 232 translates to [latex]=232[/latex] because “is” is associated with equals.
- Write an equation: [latex]5x-28=232[/latex]

[latex]\begin{array}{r}5x-28=232\\5x=260\\x=52\,\,\,\end{array}[/latex]

[latex]\begin{array}{r}5\left(52\right)-28=232\\5\left(52\right)=260\\260=260\end{array}[/latex].

We apply the idea of consecutive integers to solving a word problem in the following example.

The sum of three consecutive integers is 93. What are the integers?

- Read and understand: We are looking for three numbers, and we know they are consecutive integers.
- Constants and Variables: 93 is a constant. The first integer we will call x . Second: [latex]x+1[/latex] Third: [latex]x+2[/latex]
- Translate: The sum of three consecutive integers translates to [latex]x+\left(x+1\right)+\left(x+2\right)[/latex], based on how we defined the first, second, and third integers. Notice how we placed parentheses around the second and third integers. This is just to make each integer more distinct. is 93 translates to [latex]=93[/latex] because is is associated with equals.
- Write an equation: [latex]x+\left(x+1\right)+\left(x+2\right)=93[/latex]

Combine like terms, simplify, and solve.

## Distance, Rate, and Time

[latex]d=rt\\\frac{d}{r}=t[/latex]

Likewise, if we want to find rate, we can isolate r using division:

[latex]d=rt\\\frac{d}{t}=r[/latex]

What was each runner’s rate for their record-setting runs?

By the time Johnson had finished, how many more miles did Trason have to run?

How much further could Johnson have run if he had run as long as Trason?

What was each runner’s time for running one mile?

(rounded to two decimal places)

We can fill in our table with this information.

Now that we know each runner’s rate we can answer the second question.

Here is the table we created for reference:

[latex]50\text{ miles }-45.82\text{ miles }=1.48\text{ miles }[/latex]

Read and Understand: The word further implies we are looking for a distance.

Johnson would have run 54.6 miles, so that’s 4.6 more miles than than he ran for the race.

Now we will tackle the last question where we are asked to find a time for each runner.

We will need to divide to isolate time.

## Simple Interest

Below is a table showing the result of solving for each individual variable in the formula.

Substitute in the values given for the Principal, Rate, and Time.

Rewrite 0.7% as the decimal 0.007, then multiply.

[latex]\begin{array}{l}I=2000\cdot 0.007\cdot 24\\I=336\end{array}[/latex]

Add the interest and the original principal amount to get the total amount in her account.

[latex] \displaystyle 2000+336=2336[/latex]

She has $2336 after 24 months.

Alex invests $600 at 3.25% monthly interest for 3 years. What amount of interest has Alex earned?

[latex]{T}=3\text{ years }\cdot12\frac{\text{ months }}{ year }=36\text{ months }[/latex]

Substitute the given values into the formula.

[latex]{T}=10\text{ years }\cdot12\frac{\text{ months }}{ year }=120\text{ months }[/latex]

Substitute the given values into the formula

[latex]I=1000\,\cdot \,0.009\,\cdot \,120\\I=1080[/latex]

Our solution checks out. Jodi invested $1000.

## Further Applications of Linear Equations

Here is what you have written down:

Perimeter = 16.4 feet Length = 4.7 feet

Read and Understand: We know perimeter = 16.4 feet and length = 4.7 feet, and we want to find width.

Define the known and unknown dimensions:

First we will substitute the dimensions we know into the formula for perimeter:

Then we will isolate w to find the unknown width.

This video shows a similar garden box problem.

Isolate the term containing the variable, w, from the formula for the perimeter of a rectangle :

[latex]{P}=2\left({L}\right)+2\left({W}\right)[/latex].

First, isolate the term with w by subtracting 2 l from both sides of the equation.

Next, clear the coefficient of w by dividing both sides of the equation by 2.

You can rewrite the equation so the isolated variable is on the left side.

[latex]w=\frac{p-2l}{2}[/latex]

The area of a triangle is given by [latex] A=\frac{1}{2}bh[/latex] where

A = area b = the length of the base h = the height of the triangle

Find the base ( b) of a triangle with an area ( A ) of 20 square feet and a height ( h) of 8 feet.

Use the formula for the area of a triangle, [latex] {A}=\frac{{1}}{{2}}{bh}[/latex] .

Substitute the given lengths into the formula and solve for b.

The base of the triangle measures 5 feet.

Use the multiplication and division properties of equality to isolate the variable b .

Write the equation with the desired variable on the left-hand side as a matter of convention:

Use the multiplication and division properties of equality to isolate the variable h .

## Temperature

[latex]C=\left(F-32\right)\cdot \frac{5}{9}[/latex]

Substitute the given temperature in[latex]{}^{\circ}{C}[/latex] into the conversion formula:

[latex]12=\left(F-32\right)\cdot \frac{5}{9}[/latex]

Isolate the variable F to obtain the equivalent temperature.

Solve the formula shown below for converting from the Fahrenheit scale to the Celsius scale for F.

[latex]\begin{array}{l}\frac{9}{5}\,C+32=F-32+32\\\\\frac{9}{5}\,C+32=F\end{array}[/latex]

[latex]F=\frac{9}{5}C+32[/latex]

## Think About It

Next, isolate the variable h by dividing both sides of the equation by [latex]2\pi r[/latex].

- "Ann Trason." Wikipedia. Accessed May 05, 2016. https://en.wikipedia.org/wiki/Ann_Trason . ↵
- "American River 50 Mile Endurance Run." Wikipedia. Accessed May 05, 2016. https://en.wikipedia.org/wiki/American_River_50_Mile_Endurance_Run . ↵
- Writing Algebraic Expressions. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/uD_V5t-6Kzs . License : CC BY: Attribution
- Write and Solve a Linear Equations to Solve a Number Problem (1). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/izIIqOztUyI . License : CC BY: Attribution
- Write and Solve a Linear Equations to Solve a Number Problem (Consecutive Integers). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/S5HZy3jKodg . License : CC BY: Attribution
- Screenshot: Ann Trason Trail Running. Authored by : Lumen Learning. License : CC BY: Attribution
- Problem Solving Using Distance, Rate, Time (Running). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/3WLp5mY1FhU . License : CC BY: Attribution
- Simple Interest - Determine Account Balance (Monthly Interest). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/XkGgEEMR_00 . License : CC BY: Attribution
- Simple Interest - Determine Interest Balance (Monthly Interest). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/mRV5ljj32Rg . License : CC BY: Attribution
- Simple Interest - Determine Principal Balance (Monthly Interest). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/vbMqN6lVoOM . License : CC BY: Attribution
- Find the Width of a Rectangle Given the Perimeter / Literal Equation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/jlxPgKQfhQs . License : CC BY: Attribution
- Find the Base of a Triangle Given Area / Literal Equation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/VQZQvJ3rXYg . License : CC BY: Attribution
- Convert Celsius to Fahrenheit / Literal Equation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/DRydX8V-JwY . License : CC BY: Attribution
- Ann Trason. Provided by : Wikipedia. Located at : https://en.wikipedia.org/wiki/Ann_Trason . License : CC BY-SA: Attribution-ShareAlike
- American River 50 Mile Endurance Run. Provided by : Wikipedia. Located at : https://en.wikipedia.org/wiki/American_River_50_Mile_Endurance_Run . License : CC BY-SA: Attribution-ShareAlike

## Solving Equations

An equation says that two things are equal. It will have an equals sign "=" like this:

what is on the left (x − 2) equals what is on the right (4)

So an equation is like a statement " this equals that "

## What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

## Example: x − 2 = 4

When we put 6 in place of x we get:

How about other values for x ?

- For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
- For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

## More Than One Solution

There can be more than one solution.

## Example: (x−3)(x−2) = 0

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

## Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

## Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

## How to Solve an Equation

There is no "one perfect way" to solve all equations.

## A Useful Goal

But we often get success when our goal is to end up with:

## Example: Solve 3x−6 = 9

and a short calculation reveals that x = 5

## Like a Puzzle

Here are some things we can do:

- Add or Subtract the same value from both sides
- Clear out any fractions by Multiplying every term by the bottom parts
- Divide every term by the same nonzero value
- Combine Like Terms
- Expanding (the opposite of factoring) may also help
- Recognizing a pattern, such as the difference of squares
- Sometimes we can apply a function to both sides (e.g. square both sides)

## Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

## Special Equations

There are special ways of solving some types of equations. Learn how to ...

## Check Your Solutions

You should always check that your "solution" really is a solution.

## How To Check

Take the solution(s) and put them in the original equation to see if they really work.

## Example: solve for x:

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

Which can be solved by having x=3

Let us check x=3 using the original question:

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

"Solving" only gives us possible solutions, they need to be checked!

- Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
- Show all the steps , so it can be checked later (by you or someone else)

- Solve equations and inequalities
- Simplify expressions
- Factor polynomials
- Graph equations and inequalities
- Advanced solvers
- All solvers
- Arithmetics
- Determinant
- Percentages
- Scientific Notation
- Inequalities

Go to the Equation Plotting page

## Introduction to Equations

Example 1 Consider the equation 2x-1 = x+2

## Math Topics

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

## Unit 2: Lesson 1

- Why we do the same thing to both sides: Variable on both sides
- Intro to equations with variables on both sides
- Equations with variables on both sides: 20-7x=6x-6
- Equation with variables on both sides: fractions
- Equation with the variable in the denominator

## Equations with variables on both sides

- Your answer should be
- an integer, like 6 6 6 6
- a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5
- a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4
- a mixed number, like 1 3 / 4 1\ 3/4 1 3 / 4 1, space, 3, slash, 4
- an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75
- a multiple of pi, like 12 pi 12\ \text{pi} 1 2 pi 12, space, start text, p, i, end text or 2 / 3 pi 2/3\ \text{pi} 2 / 3 pi 2, slash, 3, space, start text, p, i, end text

## Word Problems on Linear Equations

## Step-by-step application of linear equations to solve practical word problems:

1. The sum of two numbers is 25. One of the numbers exceeds the other by 9. Find the numbers.

More solved examples with detailed explanation on the word problems on linear equations.

How to Solve Linear Equations?

Problems on Linear Equations in One Variable

Word Problems on Linear Equations in One Variable

Practice Test on Linear Equations

Practice Test on Word Problems on Linear Equations

Worksheet on Word Problems on Linear Equation

7th Grade Math Problems 8th Grade Math Practice From Word Problems on Linear Equations to HOME PAGE

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## IMAGES

## VIDEO

## COMMENTS

The four steps for solving an equation include the combination of like terms, the isolation of terms containing variables, the isolation of the variable and the substitution of the answer into the original equation to check the answer.

The six steps of problem solving involve problem definition, problem analysis, developing possible solutions, selecting a solution, implementing the solution and evaluating the outcome. Problem solving models are used to address issues that...

Whether you love math or suffer through every single problem, there are plenty of resources to help you solve math equations. Skip the tutor and log on to load these awesome websites for a fantastic free equation solver or simply to find an...

Problem Solving with Equations. Watch later. Share. Copy link. Info. Shopping. Tap to unmute. If playback doesn't begin shortly

Read and understand the problem. · Determine the constants and variables in the problem. · Translate words into algebraic expressions and equations. · Write an

Practice Solving Equations. Names: Solve each: 1). 2). 3). 4). 5). 6). 7). 8). Page 2. 9). 10) Solve for x: 11). 12)

Solving Equations · What is an Equation? · What is a Solution? · More Than One Solution · Solutions Everywhere! · How to Solve an Equation · Like a Puzzle · Special

SOLVING EQUATIONS · x - 4 = 10 Solution · 2x - 4 = 10 Solution · 5x - 6 = 3x - 8 Solution · tex2html_wrap_inline575 Solution · tex2html_wrap_inline577 Solution · 2(3x

If an equation is true after the variable has been replaced by a specific number, then the number is called a solution of the equation and is said to satisfy it

(c) Form an equation in x and solve it to work out Sarahʼs age.

Problem. Solve for f f ff. − f + 2 + 4 f = 8 − 3 f -f+2+4f=8-3f −f+2+4f=8−3fminus, f, plus, 2, plus, 4, f, equals, 8, minus, 3, f. f = f = f=f, equals.

Writing an equation to model a real-world problem is often easier when you take the information given in the problem and express it in verbal

Word Problems on Linear Equations · Then the other number = x + 9. Let the number be x. · 2.The difference between the two numbers is 48. · 3. The length of a

There are simple problems that involve linear equations. For example, the sum of 35 and a number is 72. What is the number? The thing we don't