Writing Equations to Represent Relationships

1.2.3 Writing Equations to Represent Relationships

1.2.3 • Writing Equations to Represent Relationships

Activity

Write an equation to represent each situation.

Blueberries are $4.99 a pound. Diego buys $b$ pounds of blueberries and pays $14.95.

Compare your answer: $4.99 \cdot b = 14.95$

Blueberries are $4.99 a pound. Jada buys $p$ pounds of blueberries and pays $c$ dollars.

Compare your answer: $4.99 \cdot p = c$

Blueberries are $d$ dollars a pound. Lin buys $q$ pounds of blueberries and pays $t$ dollars.

Compare your answer: $d \cdot q = t$

Noah earned $n$ dollars over the summer. Mai earned $275, which is $45 more than Noah did.

Compare your answer: $n + 45 = 275$ or $275 - 45 = n$ (or equivalent)

Noah earned $v$ dollars over the summer. Mai earned $m$ dollars, which is 45 dollars more than Noah did.

Compare your answer: $v + 45 = m$ or $m - 45 = v$ (or equivalent)

Noah earned $w$ dollars over the summer. Mai earned $x$ dollars, which is $y$ dollars more than Noah did.

Compare your answer: $w + y = x$ or $x - y = w$ (or equivalent)

How are the equations you wrote for the blueberry purchases like the equations you wrote for Mai’s and Noah’s summer earnings? How are they different?

Compare your answer:

The two sets of equations are alike in that in each set:

Each equation involves three quantities.
The first equation has two known quantities, the second has one known quantity, and the last one has no known quantities.

They are different in that:

The three quantities in each set are different. In the first set, they are unit price, pounds of blueberries, and total cost. In the second set, they are Noah’s earnings, Mai’s earnings, and the difference between the two.
In the first set, the relationship involves multiplication. In the second, it involves addition (or subtraction).

Self Check

Max and Jules are renting a car for their 3-day trip. The rental company charges $50 per day and $0.40 per mile driven. Which is a possible equation for their cost of the rental car?

$C=50(3)+0.40m$
$C=50(3)+40(3)+0.40m$
$C=50m+40(3)$
$C=50(3)+40m$

Additional Resources

Writing an Equation to Represent a Real-World Problems

Here are the steps to writing an equation to represent a real-world scenario:

Step 1 - Read the problem.

Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.

Step 3 - Write a sentence using the relationship among the values.

Step 4 - Translate the sentence into an equation.

Example

Write the following statement algebraically using symbols: Peanuts cost $3.59 per pound. Max buys $p$ pounds of peanuts and pays $10.77.

Step 1 - Read the problem.

Peanuts cost $3.59 per pound. Max buys $p$ pounds of peanuts and pays $10.77.

Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.

$3.59 = cost per pound of peanuts

$10.77 = total cost of peanuts bought

$p$ = number of pounds of peanuts bought

Step 3 - Write a sentence using the relationship among the values.

The cost per pound times the number of pounds bought equals the total cost of peanuts

Step 4 - Translate the sentence into an equation.

$3.59 (p) = 10.77$

Try it

Writing an Equation to Represent a Real-World Problem

Translate the following scenario into an equation:

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

After reading the problem, what are some of the known variables identified in the description?

Together the husband and wife earn $110,000 a year.

The wife earns $16,000 less than twice what her husband earns.

Identify a variable and describe what it represents for this scenario.

$h$ = the amount the husband earns.

$2 h - 16, 000$ = the amount the wife earns.

What sentence can be used to describe the relationship among the values?

Together the husband and wife earn $110,000.

What equation can be used to represent the scenario?

$h + 2 h - 16, 000 = $ 110, 000$

Video: Writing an Equation to Represent a Real-World Problem

Watch the following video to learn more about how to write an equation to represent a real-world problem.

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