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4. Equations & Inequalities
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United States of AmericaVirginia
Grade 7

4.03 Solve problems with two-step equations

Lesson
Practice

Solve problems with two-step equations

Once we have translated a verbal situation into an equation , we can use the equation to solve the problem by finding the unkown value of the variable .

When working to solve problems, always be sure to:

  • Read the question at least twice and find keywords.
  • Be clear what your variable is representing, for example instead of saying that "x is chickens" we could say "x is the number of chickens" or "x is the weight of each chicken."
  • Check your answer to see if it is reasonable and then check it by substituting into your equation.

Let's consider the following scenario:

You bought a chocolate bar and two bags of chips. The price of the chocolate bar was clearly marked as \$3 but there was no price on the chips. You spent a total of \$7. How much did each bag of chips cost?

The unknown value in this scenario is the cost of one bag of chips. We can represent this scenario and calculate the cost of the chips using a pictorial model like this:

The image shows 2 chips plus 3 dollars equals to 7 dollars, 2 chips equals 4 dollars, and 1 chip equals 2 dollars.

We can also translate the scenario into an equation. The cost of the bag of chips is unknown, so we will use c to represent this. This scenario can be translated into the equation 2c+3=7. Now that the scenario is translated, we can solve the equation.

\displaystyle 2c+3\displaystyle =\displaystyle 7Translate the equation
\displaystyle 2c\displaystyle =\displaystyle 4Subtract 4 from both sides
\displaystyle c\displaystyle =\displaystyle 2Divide both sides by 2

Each bag of chips costs \$2.

We can verify our solution by substiuting 2 into the equation for c.

\displaystyle 2(2)+3\displaystyle =\displaystyle 7Substitute c=2
\displaystyle 4+3\displaystyle =\displaystyle 7Evaluate the multiplication
\displaystyle 7\displaystyle =\displaystyle 7Evaluate the addition

c = 2 is a solution to the equation 2c + 3 = 7 because 2(2) + 3 = 7.

The solution of c=2 means that each bag of chips costs \$2. Given the context, this answer makes sense.

Examples

Example 1

The sum of 7 and 8x is equal to 47.

Construct the equation and find the value of x.

Worked Solution
Create a strategy

Translate the keywords into mathematical operations to set-up the equation.

Apply the idea
\displaystyle 7 +8x\displaystyle =\displaystyle 47Write the equation
\displaystyle 8x+7-7\displaystyle =\displaystyle 47-7Subtract 7 from both sides
\displaystyle 8x\displaystyle =\displaystyle 40Evaluate the subtraction
\displaystyle \dfrac{8x}{8}\displaystyle =\displaystyle \dfrac{40}{8}Divide both sides by 8
\displaystyle x\displaystyle =\displaystyle 5Evaluate the division
Reflect and check

We can substitute 5 into the originial equation to verify our answer.

\displaystyle 7+8(5)\displaystyle =\displaystyle 47Substitute x=5
\displaystyle 7+40\displaystyle =\displaystyle 47Evaluate the multiplication
\displaystyle 47\displaystyle =\displaystyle 47Evaluate the addition

Example 2

Sally and Eileen do some fundraising for their sporting team. Together, they raised \$ 600. If Sally raised \$272 more than Eileen, and Eileen raised \$ p:

a

Write and solve an equation in terms of p that represents the relationship between the different amounts.

Worked Solution
Create a strategy

Translate the keywords into mathematical operations to set up the equation.

Apply the idea

p represents the amount of money Eileen raised and p+272 is the amount Sally raised.

Adding these amounts will give the total \$600 raised.

\displaystyle p + p + 272\displaystyle =\displaystyle 600Write the equation
\displaystyle 2p + 272 =\displaystyle =\displaystyle 600Combine like terms
\displaystyle 2p + 272 - 272\displaystyle =\displaystyle 600 - 272Subtract 272 from both sides
\displaystyle 2p\displaystyle =\displaystyle 328Evaluate the subtraction
\displaystyle \dfrac{2p}{2}\displaystyle =\displaystyle \dfrac{328}{2}Divide both sides by 2
\displaystyle p\displaystyle =\displaystyle 164Evaluate the division
Reflect and check

We can substitute \$164 back into the equation to see if our solution makes the equation true.

\displaystyle 2(164)+272\displaystyle =\displaystyle 600Substitute the answer into the original equation
\displaystyle 328+272\displaystyle =\displaystyle 600Evaluate the multiplication
\displaystyle 600\displaystyle =\displaystyle 600Evaluate the addition
b

Now, calculate how much Sally raised.

Worked Solution
Create a strategy

We know that Sally raised \$272 more than Eileen.

Apply the idea

Eileen raised \$164 and Sally raised \$272 more than that.

\displaystyle \text{Amount Sally raised}\displaystyle =\displaystyle p+272Write an equation for the amount Sally raised
\displaystyle \text{Amount Sally raised}\displaystyle =\displaystyle 164 + 272Substitute p=272
\displaystyle \text{Amount Sally raised}\displaystyle =\displaystyle 436Evaluate the addition

Sally raised \$ 436.

Reflect and check

Check your solutions by adding the amount that we found Eileen raised to the amount that Sally raised to see if it equals the total amount raised of \$600.

\displaystyle 436+272\displaystyle =\displaystyle 600
\displaystyle 600\displaystyle =\displaystyle 600

This confirms our answer.

Example 3

Consider the following equation.

7.50h+25=115

a

Create a real world scenario that could be represented by the equation.

Worked Solution
Create a strategy

We know that 7.50h represents a quantity that will occur more than once, where the 25 term only occurs once.

Apply the idea

Here is one possible scenario:

A rental service charges \$7.50 per hour plus an additional \$25 one-time registration fee.

You have a budget of \$115. This equation can be used to determine how many hours you can afford to use the rental service.

Let h be the number of hours a rental service is used.

Reflect and check

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

b

Solve the equation and explain the answer in context to the scenario you created in part a.

Worked Solution
Create a strategy

Use inverse operations to isolate the variable.

Apply the idea
\displaystyle 7.50h+25\displaystyle =\displaystyle 115Write the equation
\displaystyle 7.50h+25-25\displaystyle =\displaystyle 115-25Subtract 25 from both sides
\displaystyle 7.50h\displaystyle =\displaystyle 90Evaluate the subtraction
\displaystyle \dfrac{7.50h}{7.50}\displaystyle =\displaystyle \dfrac{90}{7.50}Divide both sides by 7.50
\displaystyle h\displaystyle =\displaystyle 12Evaluate the division

This means that someone can use the rental service for 12 hours and will pay \$115

Reflect and check

We can substitute 12 back into the equation to see if our solution makes the equation true.

\displaystyle 7.50(12)+25\displaystyle =\displaystyle 115Substitute 12 into the original equation
\displaystyle 90+25\displaystyle =\displaystyle 115Evaluate the multiplication
\displaystyle 115\displaystyle =\displaystyle 115Evaluate the addition

This confirms 12 is a solution to the equation.

Idea summary

When working to solve problems, always be sure to:

  • Read the question at least twice and find keywords.
  • Be clear what your variable is representing, for example instead of saying that "x is chickens" we could say "x is the number of chickens" or "x is the weight of each chicken."
  • Check your answer to see if it is reasonable and then check it by substituting into your equation.

Outcomes

7.PFA.3

The student will write and solve two-step linear equations in one variable, including problems in context, that require the solution of a two-step linear equation in one variable.

7.PFA.3d

Write a two-step linear equation in one variable to represent a verbal situation, including those in context.

7.PFA.3e

Create a verbal situation in context given a two-step linear equation in one variable.

7.PFA.3f

Solve problems in context that require the solution of a two-step linear equation.

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