1.11: Solve and Check SingleVariable Equations Using Mental Math and Substitution
Fitzgerald Middle School had a wonderful cookie fundraiser. Within the first month, the students were selling out the inventory every single day. They were having such a difficult time keeping up, that they hired the home economics classes to help them with the baking. So some of the money went to the students who agreed to bake, but the rest of it went into the student council.
By midterms, the students calculated that after paying the home economics students, that they were still averaging $60.00 profit per week. By midterms, they had collected $540.00 total.
“How many weeks did it take us to make that much?” Jesse asked at lunch one day.
“I don’t know, but I am sure that we can make $1000.00 by end of the semester,” Tracy said smiling.
“How can you be so sure?”
“You just do the math. First, we can write an equation and solve it to figure out how long it took us to make the $540.00. Then double it for the end of the semester,” Tracy explained.
Do you understand how Tracy figured this out? Pay attention and you will be able to solve this dilemma by the end of the Concept.
Guidance
An equation includes groups of numbers, symbols, and variables. However, equations also include an equals sign.
The key thing to remember about an equation is that the quantity on one side of the equals must be the same as the quantity on the other side of the equals.
There are different ways to solve an equation.
When you solve an equation, you are solving to determine the value of the variable. If you choose the correct value for the variable, then the equation will be a true statement. Let’s look solving an equation without a variable.
\begin{align*}4+16=20\end{align*}
We can look at the quantity on the left side of the equation first. It is equal to 20. The right side of the equation is also 20. This is a true statement.
An equation must always make a true statement. We can say that this is a balanced equation.
What if this equation had a variable in place of one of the numbers?
\begin{align*}x+16=20\end{align*}
Now we have a puzzle to solve. We can start by thinking about what number plus sixteen is equal to 20. We know that four plus sixteen is equal to 20. So, the value of \begin{align*}x\end{align*}
We write the answer to an equation in a particular way.
The answer is that \begin{align*}x=4\end{align*}
Think a little deeper about how you solved this. If you think about it you probably subtracted 20 – 16 in your head. This is called using an inverse operation. The inverse operation is the opposite operation. We can use inverse operations to solve equations.
Here is another one.
\begin{align*}4x=12\end{align*}
Here we have a multiplication problem. We can ask ourselves, what number times four is equal to 12? The answer is 3.
\begin{align*}x=3\end{align*}
We could also use the inverse operation to solve this. Twelve divided by four is three. Our answer is the same and both methods can be completed using mental math.
You can also check an answer by substituting it back into the original problem.
\begin{align*}5x+3=18\end{align*}
After solving this equation using mental math, we figure out the value of the variable is three. We can check this answer by substituting the value of the variable back into the original equation. Then we simplify it. If the equation makes a true statement, then we know that we have the correct answer.
\begin{align*}5(3) + 3 &= 18\\
15 + 3 &= 18\\
18 &= 18\end{align*}
This is a true statement so our work is accurate.
You can solve these equations by using mental math.
Example A
\begin{align*}\frac{x}{2}= 7\end{align*}
Solution: \begin{align*}x=14\end{align*}
Example B
\begin{align*}\frac{22}{x}=11\end{align*}
Solution: \begin{align*}x = 2\end{align*}
Example C
\begin{align*}8x = 64\end{align*}
Solution: \begin{align*}x = 8\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
To work on this problem, first we need to write an equation. Let’s look at what we know.
We know that the students averaged $60.00 profit per week.
We know that their gross profit was $540.00.
We need to know how many weeks it took them to earn that. Our variable is the number of weeks, \begin{align*}w\end{align*}
Here is our equation.
\begin{align*}60w=540\end{align*}
We can solve this using mental math.
It took the students 9 weeks to earn the money.
Vocabulary
 Equation
 a group of numbers, operations and variables where the quantity on one side of the equal sign is the same as the quantity on the other side of the equal sign.
 Inverse Operation
 the opposite operation. Equation can often be solved by using an inverse operation.
Guided Practice
Here is one for you to try on your own.
\begin{align*}5x+3=18\end{align*}
Solution
Let’s break down this equation by using saying it to ourselves.
“Five times some number plus three is equal to eighteen.” Now you can think through the five times table for an answer that makes sense.
5, 10, 15, 20
15 makes sense so that would make the variable equal to three since five times three is fifteen.
\begin{align*}x=3\end{align*}
This is our answer.
Video Review
Practice
Directions: Solve each equation using mental math. Be sure to check each answer by substituting your solution back into the original problem. Then simplify to be sure that your equation is balanced.

\begin{align*}x+4=22\end{align*}
x+4=22  \begin{align*}y+8=30\end{align*}
 \begin{align*}x19=40\end{align*}
 \begin{align*}12x=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*}8y2=54\end{align*}
 \begin{align*}3b+12=30\end{align*}
 \begin{align*}9y7=65\end{align*}
 \begin{align*}12a5=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*}5a12=33\end{align*}
 \begin{align*}7b9=33\end{align*}
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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
Here you'll solve and check singlevariable equations using mental math and substitution.