Topics
6. Equations & Inequalities
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United States of America
Grade 7

6.03 Problem solving with equations

Lesson
Practice

Introduction

Translating a word problem into an equation is a skill we will use often in our mathematics studies. It is always helpful to identify keywords and know how they might translate to math.

We have had some practice with this when  writing equations  , but now we want to put it all together and solve the problems that we've written equations to represent.

Problem solving with equations

When translating word problems into equations, we may need to use a variable to represent the unknown quantity we are looking for, we will often use x or n.

It is also helpful to know some keywords or phrases that point to certain parts of the equation.

AdditionSubtractionMultiplicationDivisionEqual
plusminustimesdivided byis/are
the sum ofthe difference ofthe product ofthe quotient ofequals
increased bydecreased bymultiplied byseparated into equal partsamounts to
totalfewer thanofsplittotals
more thanless thantwiceequally shared
added tosubtracted fromgroups/lots of

Once we have translated to a mathematical sentence, we can solve the problem. Solving the problem means solving for the unknown, or 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.

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 40Divide both sides of the equation by 8 to find x
\displaystyle x\displaystyle =\displaystyle 5Evaluate

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 an equation in p that represents the relationship between the different amounts, and solve for p.

Worked Solution
Create a strategy

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

Apply the idea
\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 328Divide both side by 2p
\displaystyle p\displaystyle =\displaystyle 164Evaluate
Reflect and check

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

b

Now, calculate how much Sally raised.

Worked Solution
Create a strategy

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

Apply the idea
\displaystyle \text{Amount raised}\displaystyle =\displaystyle 164 + 272Add the amount that Eileen raised to \$272
\displaystyle \text{Amount raised}\displaystyle =\displaystyle 436Evaluate

So 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.

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.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.

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