Identifying Equivalent Expressions in Context | Grade 7 Math | LevelKite

When two people describe the same situation with math, their work often looks different. One person adds up each part. Another groups things first and then multiplies.

Two words come up all through this lesson. A variable is a letter that stands for a number that can change. An expression is a calculation written with numbers, variables and operation signs, such as $12n + 3n$ .

You have already rewritten an expression in a new form to solve a problem. Now you will judge whether two different-looking expressions always give the same amount, and you will see what each form tells you about the situation.

A class is going to a science museum. Each student pays 12 dollars for a ticket and 3 dollars for a snack. Let $n$ be the number of students who go. Because $n$ counts people, it is a whole number: 0, 1, 2, 3 and so on.

Maya thinks about tickets and snacks separately. The tickets cost $12n$ dollars, which means 12 times $n$ , and the snacks cost $3n$ dollars. She writes the total cost as $12n + 3n$ dollars.

Leo thinks about one student first. Each student pays $12 + 3 = 15$ dollars, so he writes the total cost as $15n$ dollars.

Maya and Leo are counting the same money in two ways. If both are right, their expressions must give the same total for every number of students.

Total cost in dollars for some numbers of students

Number of students	Maya's total in dollars	Leo's total in dollars
$n = 1$	$12(1) + 3(1) = 12 + 3 = 15$	$15(1) = 15$
$n = 2$	$12(2) + 3(2) = 24 + 6 = 30$	$15(2) = 30$
$n = 5$	$12(5) + 3(5) = 60 + 15 = 75$	$15(5) = 75$
$n = 10$	$12(10) + 3(10) = 120 + 30 = 150$	$15(10) = 150$

The two totals match in every row. That is good evidence, but it is not proof. A table shows only a few values of $n$ , and the totals have to match for every value of $n$ .

To be sure, you need a reason that works for all numbers at once, not one number at a time.

Key concept

Equivalent expressions

Two expressions are equivalent when they have the same value for every value of the variable. Matching for only some values is not enough.

In a problem, equivalent expressions always give the same amount, so they describe the same quantity in different ways.

To show that two expressions are equivalent, rewrite one into the other using a rule that is true for all numbers. Then the two forms are equal for every value of the variable.
To show that two expressions are not equivalent, find one value of the variable where they give different results.

The rule you will use most is the distributive property. It says that multiplying a number by a sum gives the same result as multiplying the number by each part of the sum and then adding the results. In symbols, $a(b + c) = ab + ac$ . It works the same way for subtraction: $a(b - c) = ab - ac$ .

Both statements are true for all numbers $a$ , $b$ and $c$ , whether they are positive or negative, whole numbers, fractions or decimals.

You can also use the property in reverse. The order in which you multiply two numbers never changes the result. So when the same number multiplies two parts, you can add the other two numbers first and then multiply once. For example, $12n + 3n = (12 + 3)n$ for every number $n$ .

Worked example

Two ways to total a class trip

Each student on a class trip pays 12 dollars for a ticket and 3 dollars for a snack. Let $n$ be the number of students, a whole number. Maya writes the total cost in dollars as $12n + 3n$ . Leo writes it as $15n$ . Are the two expressions equivalent?

Start with what Maya's expression means. The part 12n is the cost of n tickets in dollars, and the part 3n is the cost of n snacks in dollars. Adding the two parts gives the total cost.
$12n + 3n$
The same number, n, multiplies both parts. Use the distributive property in reverse: add 12 and 3 first, then multiply by n. The property is true for all numbers, so this line is true for every value of n.
$12n + 3n = (12 + 3)n$
Add inside the parentheses. Since 12 + 3 and 15 are the same number, writing 15 does not change the value of the expression for any value of n.
$(12 + 3)n = 15n$
Put the steps together. Each line is true for every value of n, so Maya's expression and Leo's expression are equal for every value of n. That covers every whole number of students, not only the four numbers in the table.
$12n + 3n = 15n$

Answer

Yes. $12n + 3n$ and $15n$ are equivalent because they are equal for every value of $n$ . Both give the total cost in dollars, and $15n$ shows that each student pays 15 dollars in all.

Worked example

Three lunches, three expressions

Three friends each buy one sandwich and one drink. A sandwich costs $s$ dollars, where $s$ is greater than 0, and a drink costs 2 dollars. Which of these expressions give the total cost in dollars for every value of $s$ : $3(s + 2)$ , $3s + 6$ , $3s + 2$ ?

Build an expression from the story. One friend pays s dollars for a sandwich and 2 dollars for a drink, which is s + 2 dollars. Three friends pay 3 times that amount, so the first expression fits the story.
$3(s + 2)$
Use the distributive property: multiply 3 by each part of the sum. The property is true for all numbers, so the second expression equals the first for every value of s. In the story, 3s dollars pays for 3 sandwiches and 6 dollars pays for 3 drinks.
$3(s + 2) = 3 \cdot s + 3 \cdot 2 = 3s + 6$
Test the third expression with one value, s = 5. The two results are different. Equivalent expressions are equal for every value of s, so the third expression is not equivalent to the first. It pays for only one drink.
$3(5 + 2) = 3(7) = 21 \quad \text{but} \quad 3(5) + 2 = 15 + 2 = 17$

Answer

$3(s + 2)$ and $3s + 6$ both give the total cost in dollars. They are equivalent because they are equal for every value of $s$ . $3s + 2$ is not equivalent: it counts one 2-dollar drink instead of three.

Worked example

Taking 20% off a price

A store takes 20% off the regular price of every jacket. Let $p$ be the regular price of a jacket in dollars, where $p$ is greater than 0. Which of these expressions give the sale price in dollars for every value of $p$ : $p - 0.2p$ , $0.8p$ , $p - 0.2$ ?

Build an expression from the story. 20% means 20 out of 100, and 20 divided by 100 is 0.2. So the discount is 0.2 times the regular price, or 0.2p dollars. The sale price is the regular price minus the discount.
$p - 0.2p$
Write p as 1p, because multiplying by 1 does not change a number. Now p multiplies both parts, so use the distributive property in reverse: subtract 0.2 from 1 first, then multiply by p. The property is true for subtraction and for all numbers, so this line is true for every value of p. The minus sign stays, because you still take the discount away.
$p - 0.2p = 1p - 0.2p = (1 - 0.2)p$
Subtract inside the parentheses: 1 minus 0.2 is 0.8. Both lines are true for every value of p, so the second expression is equivalent to the first. In the story, 0.8 is 80 out of 100, so taking 20% off means paying 80% of the regular price.
$(1 - 0.2)p = 0.8p$
Test the third expression with one value, p = 50. The two results are different. Equivalent expressions are equal for every value of p, so the third expression is not equivalent to the first. It takes off 0.2 dollars, which is 20 cents, instead of 20% of the price.
$50 - 0.2(50) = 50 - 10 = 40 \quad \text{but} \quad 50 - 0.2 = 49.8$

Answer

$p - 0.2p$ and $0.8p$ both give the sale price in dollars. They are equivalent because they are equal for every value of $p$ . $p - 0.2$ is not equivalent: it takes off 20 cents, not 20% of the price.

Each form shows something different about the same situation

Situation	Expression	What this form shows
Class trip	$12n + 3n$	Ticket cost plus snack cost for $n$ students
Class trip	$15n$	15 dollars for each of $n$ students
Three lunches	$3(s + 2)$	3 friends who each pay $s + 2$ dollars
Three lunches	$3s + 6$	3 sandwiches at $s$ dollars each, plus 6 dollars for 3 drinks
Jacket sale	$p - 0.2p$	Regular price minus a discount of 20% of that price
Jacket sale	$0.8p$	80% of the regular price

In each situation in the table, the two expressions are equivalent: they give the same amount for every value of the variable. Yet each form points to something different, such as the cost for each student or the percent of the price you pay. That is why it helps to rewrite an expression in a problem.

Takeaway: Two expressions are equivalent only if they are equal for every value of the variable. To show that they are, rewrite one into the other with a rule that is true for all numbers, such as the distributive property. To show that they are not, find one value where the results differ. One matching value never proves that two expressions are equivalent.