We've previously looked at how to simplify expressions . We'll now be looking at what happens when we have two expressions that are equal to each other.

Ideas

Equations
Identify solutions

Equations

An equation is a mathematical sentence or number sentence stating that two expressions are equal. That is, they have the same value. We can think of the two expressions as being balanced on scales.

A balanced scale. On the left weight is 3 x and on the right weight is x plus 4.

Here are some examples of equations:

\begin{aligned} 3x + 2 = 4 \\ 4.25 =3\frac{1}{2} + n \\ \frac{y}{3}=2y +5 \end{aligned}

Equations often contain letters or symbols that are used to represent an unknown quantity. These symbols are called variables. As seen above, variables can appear on one or both sides of an equation.

An equation is like a mathematical sentence, and we can translate a written sentence into an equation. There are some key words to look out for when writing a sentence as an equation. For example, where we see "is" or "equals", we will put an equals sign. When we see "groups of" we will multiply.

These key words will become more clear after practicing a few problems.

Examples

Example 1

Adam thought of a number, doubled it and added 5 to get 13. Write an equation that represents this scenario.

Worked Solution

Create a strategy

There is an unknown quantity so we should use a variable to represent it. Replace the words with numbers and mathematical symbols.

Apply the idea

\displaystyle n \times 2 + 5	\displaystyle =	\displaystyle 13	Write the equation
\displaystyle 2n+5	\displaystyle =	\displaystyle 13	Simplify

Example 2

What scenario could the equation 3a=2b+1 be representing?

Worked Solution

Create a strategy

We should start by stating what a \text{ and } b represent.

Apply the idea

Here is one possible scenario:

Let a be the number of apples and b be the number of bananas. The cost of 3 apples is 1 dollar more than the cost of 2 bananas.

Reflect and check

How many different scenarios can you come up with? What do all the scenarios have in common? What is different?

Example 3

Roxanne has been out picking flowers, and has 40 in total. When she returns, she puts them in 5 different vases. If she puts p flowers in each vase, rewrite the following sentence using algebra:

"There are 5 groups of p flowers, which make 40 in total."

Worked Solution

Create a strategy

The phrase "groups of" means to "multiply by" while the phrase "which makes" means that it is "equal to".

Apply the idea

\displaystyle 5p

\displaystyle =

\displaystyle 40

Write the equation

Idea summary

An equation is a mathematical sentence stating that two expressions are equal.

Equations often contain letters or symbols used to represent an unknown quantity. These are called variables.

Identify solutions

We say that a value for a variable is a solution to an equation if we can substitute it into the equation and it makes the number sentence true.

For example, if we wanted to find the solution to the equation x + 1 = 3, we want to find a value for x that makes that equation true. This statement is true when x = 2 because 2 + 1 = 3.

When we're figuring out whether a value is a solution, we need to see whether the left-hand side of the equation is the same as the right-hand side. We can think of it as a see-saw.

For example, if we have the equation x + 12 = 20, we could think of it visually as:

A set of balanced scales. On the left weight is x plus 12 and on the right weight is 20.

If 20 was removed from the left-hand side of the seesaw, it would look unbalanced like this:

A set of unbalanced scales. The left weight is x and on the right weight is 20. The right weight is lower than the left

So how do we balance the equation again? We need to remove 12 from the right-hand side as well:

A set of balanced scales. The left weight is x and the right weight is 8

So x = 8 satisfies the equation x + 12 = 20 because 8 + 12 = 20.

Examples

Example 4

There are two rectangular-shaped pools at the local aquatic center. Each pool has a length that is triple its width. Pool 1 has a perimeter of 256 feet.

Let the width of the pools be represented by w. Which of the following equations represents the perimeter of each pool in terms of w?

2w + 3w = 256

3w + 3w + 3w = 256

w + 3w = 256

w + w + 3w + 3w = 256

Worked Solution

Create a strategy

The perimeter of a rectangle is equal to the sum of its four sides and it has two sets of equal sides.

Apply the idea

\displaystyle \text{Length}	\displaystyle =	\displaystyle 3\times w	Length is triple the width
	\displaystyle =	\displaystyle 3w	Simplify
\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle \text{width + width + length + length}	Perimeter of a rectangle
\displaystyle 256	\displaystyle =	\displaystyle w+w+3w+3w	Substitute values and expressions

The answer is Option D w + w + 3w + 3w = 256.

The width of Pool 2 is 32 feet. Find its perimeter.

Worked Solution

Create a strategy

Substitute the given width value into the perimeter equation.

Apply the idea

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 32 + 32 + 3 \times 32 + 3 \times 32	Substitute the value of w
	\displaystyle =	\displaystyle 32 + 32 + 96 + 96	Evaluate multiplication
	\displaystyle =	\displaystyle 256 \text{ feet}	Evaluate

Is the width of Pool 1 also 32 feet?

Worked Solution

Create a strategy

Compare the perimeters of Pool 1 and Pool 2.

Apply the idea

Both pools have a perimeter of 256\text{ ft,} and both pools have lengths that are triple the width. So the width of Pool 1 is also 32 \text { feet}.

Example 5

We want to determine if b = 8 is the solution of 8b = 63.

Find the value of the left-hand side of the equation when b = 8.

Worked Solution

Create a strategy

Substitute the value of b into the equation.

Apply the idea

\displaystyle 8b	\displaystyle =	\displaystyle 8 \times 8	Substitute the value of b
	\displaystyle =	\displaystyle 64	Evaluate

Is b = 8 the solution of 8b = 63?

Worked Solution

Create a strategy

Recall the previous answer on part A.

Apply the idea

When b = 8, the value of the left-hand side is equal to 64 \neq 63. b = 8 is not a solution to the equation.

Idea summary

A value for a variable is a solution to an equation if we can substitute it into the equation and it makes both sides of the equation equal the same value.

Outcomes

7.EE.B.4

Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

7.EE.B.4.A

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

6.01 Identify and write equations

Introduction

Ideas

Equations

Examples

Example 1

Example 2

Example 3

Identify solutions

Examples

Example 4

Example 5

Outcomes

7.EE.B.4

7.EE.B.4.A

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