1.5: Solving Equations
Introduction
Cookie Sale Success
The cookie sale was a much bigger success than either Cameron or Tracy had even thought. 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.
In the last lesson you worked with evaluating expressions. Now let’s look at writing and solving simple equations. These are equations that you can use mental math to solve. Pay close attention and you will see this problem again at the end of the lesson.
What You Will Learn
In this lesson you will learn how to complete the following skills.
- Solve single-variable equations using mental math.
- Check suggested solutions to equations by substituting for the variable and simplifying.
- Write and solve equations using verbal models for given problem situations.
- Model and solve real-world problems using simple whole-number equations.
Teaching Time
I. Solve Single-Variable Equations Using Mental Math
Like an expression, an equation includes groups of numbers, symbols, and variables. However, equations also include equal signs. 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. In this lesson, you will learn to solve equations using mental math.
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 at an example of an equation without a variable.
Example
\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?
Example
\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*} must be four.
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.
Example
\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.
Example
\begin{align*}\frac{x}{2}= 7\end{align*}
Now we have a division problem. We have a number divided by two is equal to seven. To solve this one, we can use the inverse operation. Two times seven is fourteen.
\begin{align*}x=14\end{align*}
Example
\begin{align*}\frac{22}{x}=11\end{align*}
This one is tricky because the inverse operation won’t work. We can’t multiply 22 times 11 to get an answer that makes any sense. We want to figure out what number times 11 is 22 or 22 divided by what number is 11?
We can use the 11 times table to figure this out.
11, 22, 33, 44
Oh, the answer is 2.
\begin{align*}x=2\end{align*}
Sometimes, you will have problems that are a bit more challenging. You can still complete these using mental math. You will just have to think of two operations and not one.
Example
\begin{align*}5x+3=18\end{align*}
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*}
Sometimes looking at a problem like this one can be hard to check to see if your answer is correct. This is when checking your answer is so important.
II. Check Suggested Solutions to Equations by Substituting for the Variable and Simplifying
A solution of an equation is the value for a variable that makes an equation true. For example, five is the solution in the equation \begin{align*}2 + x = 7\end{align*} because \begin{align*}2 + 5 = 7\end{align*}. You can check to see if the solution given for an equation is correct by substituting it for the variable. Let’s look at the last example from the last section.
Example
\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.
Sometimes you will be given a value and you will need to determine if it makes a true statement.
Example
Is 40 a solution of \begin{align*}15x + 15 = 615\end{align*}?
Step 1: Substitute 40 for the variable “\begin{align*}x\end{align*}.”
\begin{align*}15x + 15 &= 615\\ 15(40) + 15 &= 615\end{align*}
Step 2: Solve the equation using the standard order of operations.
\begin{align*}15(40) + 15 &= 615\\ 600 + 15 &= 615\\ 615 &= 615\end{align*}
In this case, when 40 is substituted for the variable, both sides of the equation equal 615. Therefore, 40 is a solution of \begin{align*}15x + 15 = 615\end{align*}.
Let’s look at an example where you will also need to use the order of operations.
Example
Is 6 a solution for \begin{align*}7m + 7 - 2m = 37\end{align*}?
Recall that 6 should be substituted for the variable in the equation. Then follow the order of operations to solve.
\begin{align*}7m + 7 - 2m &= 37\\ 7(6) + 7 - 2(6) &= 37 \ (\text{Substitute the variable and multiply})\\ 42 + 7 - 12 &= 37 \ (\text{Add})\\ 49 - 12 &= 37 \ (\text{Subtract})\\ 37 &= 37\end{align*}
When 6 is substituted for the variable, both sides of the equation equal 37. Therefore, 6 is a solution for the equation \begin{align*}7m + 7 - 2m = 37\end{align*}.
6 is a solution for this equation.
III. Write and Solve Equations using Verbal Models for Given Problem Situations
Sometimes if you think of a problem in terms of words and parts it will be easier to write an equation and solve it. Writing a verbal model is similar to making a plan for solving a problem. When you write a verbal model, you are paraphrasing the information stated in the problem. After writing a verbal model, insert the values from the problem to write an equation. Then, use mental math or an inverse operation to solve it.
Let’s look at an example.
Example
Monica purchased a pair of tennis shoes on sale for $65.99. The shoes were originally $99.00. Use a verbal model to write and solve an equation to determine the amount of money Monica saved by purchasing the shoes on sale
First write a verbal model to represent the problem.
Verbal Model: \begin{align*}\text{Sale Price} + \text{Amount Saved} = \text{Original Price}\end{align*}
Let “\begin{align*}s\end{align*}” represent the amount saved.
Equation: \begin{align*}65.99 + s = 99.00\end{align*}
Solution: Recall that to solve for “\begin{align*}s\end{align*},” complete the inverse operation. Since addition is used in the equation, use subtraction to solve.
It makes sense to subtract 65.99 from 99.00.
\begin{align*}99.00 - 65.99 = \$33.01\end{align*}
This is the answer.
Example
The cost to run a thirty second commercial on prime time television is seven hundred fifty-thousand dollars. Use a verbal model to write and solve an equation to determine the cost per second.
Verbal Model: \begin{align*}\frac{\text{Total Cost}}{\text{Number of Seconds}} = \text{Cost per Second}\end{align*}
Let “\begin{align*}x\end{align*}” represent the unknown cost per second.
Equation: \begin{align*}\frac{\$750,000}{30} = x\end{align*}
Solution: To solve, divide 750,000 by 30.
\begin{align*}\frac{750,000}{30} &= x\\ 25,000 &= x\end{align*}
Now remember that we were talking about money in this problem. So our answer needs to be written as a money amount.
The answer is that is costs $25,000 per second for a thirty second commercial.
IV. Model and Solve Real-World Problems using Simple Whole-Number Equations
You can construct models to help you solve a problem. You have already learned to use a verbal model to write and solve equations to solve problems. Creating a model for a problem may also include methods such as drawing a diagram or picture or making a table or chart.
Example
The triangles below were constructed using toothpicks. Determine the number of toothpicks needed to construct twenty triangles.
As you can see, three toothpicks were needed to construct one triangle. Two more were needed to construct the second triangle. Therefore, five toothpicks were used to make two triangles. Continue to make more triangles along the row. Each time you construct a new triangle, record the number of toothpicks used on a chart.
Triangle #: | Toothpick # |
---|---|
1 | 3 |
2 | 5 |
3 | 7 |
4 | 9 |
5 | 11 |
6 | 13 |
7 | 15 |
8 | 17 |
9 | 19 |
10 | 21 |
Looking at the table, you can identify a pattern. You can see that two toothpicks are needed each time a new triangle is constructed. You can write a verbal model to express this amount.
Total Number of Toothpicks Needed = Two Times the Number of Triangles + One Toothpick
Let \begin{align*}n =\end{align*} number of triangles
Total Number of Toothpicks Needed \begin{align*}= 2n + 1\end{align*}
To determine the number of toothpicks needed to construct twenty triangles, substitute twenty for the variable.
\begin{align*}& 2n + 1\\ & 2(20) + 1\\ & 40 + 1\\ & 41\end{align*}
41 toothpicks are needed to construct twenty triangles.
Now let’s go back to the problem from the introduction and work on applying what we have learned in this lesson. With our new knowledge, we can work on solving the introductory problem.
Real-Life Example Completed
Cookie Sale Success
Here is the original problem once again. Reread it first. Then write an equation to solve for the number of weeks that it took the students to earn $540.00. There are two parts to this problem, an equation and the number of weeks.
The cookie sale was a much bigger success than either Cameron or Tracy had even thought. 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 length of the semester.” Tracy explained.
Now write an equation and solve it using mental math.
Solution to Real – Life Example
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
Here are the vocabulary words found in this lesson.
- 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.
- Verbal Model
- using words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.
Time to 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*}
- \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*}
Directions: Use what you have learned about writing verbal models to solve each equation. There are three parts to each problem.
- Alexander’s resting heart rate is 75 beats per minute. This is thirty-beats less than his heart rate after exercising. Use a verbal model. Write an equation. Then solve the equation to determine Alexander’s heart rate after exercising.
- The rectangles below were constructed using toothpicks. Four toothpicks were used to make one rectangle. Three more toothpicks were added to make a second rectangle. Determine the number of toothpicks needed to make fifteen rectangles in a row. Write a verbal model. Write an equation. Solve the equation for the solution.