# 1.11: Solve and Check Single-Variable Equations Using Mental Math and Substitution

**At Grade**Created by: CK-12

**Practice**Mental Math for Multiplication/Division Equations

Maria and her mother are shopping for a dress for Maria’s school dance. The dress Maria likes the most is $120. She asks her mom, “If I give you my cleaning money for the next six weeks, will you pay the other half?”

Maria is paid the same amount of money each week to clean her neighbor’s house. How can you use the information given here to determine how much money Maria earns each week?

In this concept, you will learn to solve and check single variable equations using mental math and substitution.

### Solving Equations Using Mental Math

An **equation** is a statement of equality of two mathematical expressions. The quantity on one side of the equals sign must have the same value as the quantity on the other side of the equals sign. For example, \begin{align*}11-6=5\end{align*}**variable** is a letter that represents a quantity.

When working with an equation involving a single variable, you are looking for the number that gives a true statement when you replace the variable by the number. This process is referred to as **solving** the equation. The number that gives the true statement is said to “satisfy” the equation and is called the **solution** or **root** of the equation.

Let’s look at an example of solving an equation with a single variable.

Solve the equation \begin{align*}x+4=12\end{align*}

You can solve this equation using mental math.

Ask yourself, “What number added to 4 gives 12 or what is the result of subtracting 4 from 12?” Either way, the solution is 8 because \begin{align*}8+4=12\end{align*}

To check your answer, return to the equation you were given to solve and substitute \begin{align*}x=8\end{align*}

\begin{align*}\begin{array}{rcl}
x+4 & = & 12\\
8+4 & = & 12
\end{array}\end{align*}

Then, add the numbers on the left side of the equals sign.

\begin{align*}\begin{array}{rcl}
8+4 & = & 12\\
12 & = & 12
\end{array}\end{align*}

The solution of \begin{align*}x=8\end{align*}

Let’s look at one more example.

Solve the equation \begin{align*}3x=18\end{align*}

You can figure out this solution by using mental math. Ask yourself, “What number times 3 gives 18 or how many times 3 divides into 18?” Either way, the solution is 6 because \begin{align*}3 \times 6 =18\end{align*}

The solution is \begin{align*}x=6\end{align*}

To check your answer, return to the equation you were given to solve and substitute \begin{align*}x=6\end{align*}

\begin{align*}\begin{array}{rcl}
3x & = & 18\\
3(6) & = & 18
\end{array}\end{align*}

Now multiply the numbers on the left side of the equals sign.

\begin{align*}18=18\end{align*}

The solution of \begin{align*}x=6\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Maria and her new dress.

You want to know how much money Maria earns each week, if six weeks worth of pay is equal to half the cost of the $120 dress.

First, divide \begin{align*}\$120.00 \div 2\end{align*}

\begin{align*}\frac{120}{2}=60\end{align*}

Next, write an equation to show that Maria is paying 6 times some amount of money \begin{align*}(x)\end{align*}

\begin{align*}6x=60\end{align*}

Then, ask yourself, “Six times what number equals 60?”

The answer is 10.

The solution is \begin{align*}x=10\end{align*}

#### Example 2

Solve the equation \begin{align*}3x-6=9\end{align*}

The equation is asking you to find 3 times what number \begin{align*}(x)\end{align*}

You can use mental math to solve this equation.

First, ask yourself, “What number subtract 6 equals 9?”

The answer is 15.

Next, ask yourself, “Three times what number \begin{align*}(x)\end{align*}

The answer is 5.

\begin{align*}x=5\end{align*}

To check your answer, return to the equation you were given to solve and substitute \begin{align*}x=5\end{align*}

\begin{align*}\begin{array}{rcl}
3x-6 & = & 9\\
3(5)-6 & = & 9
\end{array}\end{align*}

First, multiply \begin{align*}3(5)=15\end{align*}

\begin{align*}15-6=9\end{align*}

Next, subtract the numbers \begin{align*}15-6=9\end{align*} on the left side of the equals sign.

\begin{align*}9=9\end{align*}

The root is \begin{align*}x=5\end{align*}.

#### Example 3

Solve and check the equation \begin{align*}\frac{x}{2}=12\end{align*} using mental math.

First, determine what the equation is asking you to find.

The equation is asking you to find what number \begin{align*}(x)\end{align*} divided by two equals 12.

The number is 24 because \begin{align*}24 \div 2=12\end{align*}.

The root is \begin{align*}x=24\end{align*}.

CHECK:

First, substitute \begin{align*}x=24\end{align*} into the equation.

\begin{align*}\frac{24}{2}=12\end{align*}

Then, divide 24 by 2 on the left side of the equals sign.

\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ 12}{2 \overline{ ) {24 }}}\\ && \ \underline{-2}\\ && \quad 04\\ && \ \ \ \underline{-4}\\ && \quad \ \ 0 \\ \\ && 12=12 \end{array}\end{align*}

#### Example 4

Determine whether \begin{align*}x=6\end{align*} is the solution to the following equation:

\begin{align*}3x-8=12\end{align*}

First, substitute \begin{align*}x=6\end{align*} into the given equation.

\begin{align*}3(6)-8=12\end{align*}

Next, multiply \begin{align*}3(6)=18\end{align*} to clear the parenthesis.

\begin{align*}18-8=12\end{align*}

Then, subtract the numbers on the left side of the equals sign. \begin{align*}18-8=10\end{align*}

\begin{align*}10=12\end{align*}

This is not a true statement.

The answer is NOT \begin{align*}x=6\end{align*}.

### Review

Solve each equation using mental math. Be sure to check each answer by substituting your solution back into the original problem. Then simplify to see if the equation expresses a true statement.

- \begin{align*}x+4=22\end{align*}
- \begin{align*}y+8=30\end{align*}
- \begin{align*}x-19=40\end{align*}
- \begin{align*}12-x=9\end{align*}
- \begin{align*}4x=24\end{align*}
- \begin{align*}6x=36\end{align*}
- \begin{align*}9x=81\end{align*}
- \begin{align*}\frac{y}{5}=2\end{align*}
- \begin{align*}\frac{a}{8}=5\end{align*}
- \begin{align*}\frac{12}{b}=6\end{align*}
- \begin{align*}6x+3=27\end{align*}
- \begin{align*}8y-2=54\end{align*}
- \begin{align*}3b+12=30\end{align*}
- \begin{align*}9y-7=65\end{align*}
- \begin{align*}12a-5=31\end{align*}
- \begin{align*}\frac{x}{2} + 4=8\end{align*}
- \begin{align*}\frac{x}{4}+3=7\end{align*}
- \begin{align*}\frac{10}{x} +9 =14\end{align*}
- \begin{align*}5a-12=33\end{align*}
- \begin{align*}7b-9=33\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.11.

### Resources

Algebraic Expression

An expression that has numbers, operations and variables, but no equals sign.Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.### Image Attributions

In this concept, you will learn to solve and check single variable equations using mental math and substitution.

**Save or share your relevant files like activites, homework and worksheet.**

To add resources, you must be the owner of the Modality. Click Customize to make your own copy.