Lesson 15
Solving Equations with Rational Numbers
Let’s solve equations that include negative values.
15.1: Number Talk: Opposites and Reciprocals
The variables a through h all represent different numbers. Mentally find numbers that make each equation true.
\frac35 \boldcdot \frac53 = a
7 \boldcdot b = 1
c \boldcdot d = 1
\text-6 + 6 = e
11 + f = 0
g + h = 0
15.2: Match Solutions
Match each equation to a value that makes it true by dragging the answer to the corresponding equation. Be prepared to explain your reasoning.
15.3: Trip to the Mountains
The Hiking Club is on a trip to hike up a mountain.
-
The members increased their elevation 290 feet during their hike this morning. Now they are at an elevation of 450 feet.
- Explain how to find their elevation before the hike.
- Han says the equation e + 290 = 450 describes the situation. What does the variable e represent?
- Han says that he can rewrite his equation as e=450 + \text-290 to solve for e. Compare Han's strategy to your strategy for finding the beginning elevation.
-
The temperature fell 4 degrees in the last hour. Now it is 21 degrees. Write and solve an equation to find the temperature it was 1 hour ago.
-
There are 3 times as many students participating in the hiking trip this year than last year. There are 42 students on the trip this year.
- Explain how to find the number of students that came on the hiking trip last year.
- Mai says the equation 3s=42 describes the situation. What does the variable s represent?
- Mai says that she can rewrite her equation as s=\frac13 \boldcdot 42 to solve for s. Compare Mai's strategy to your strategy for finding the number of students on last year’s trip.
-
The cost of the hiking trip this year is \frac23 of the cost of last year's trip. This year's trip cost $32. Write and solve an equation to find the cost of last year's trip.
A number line is shown below. The numbers 0 and 1 are marked on the line, as are two other rational numbers a and b .

Decide which of the following numbers are positive and which are negative.
a-1
a-2
\text-b
a+b
a-b
ab+1
15.4: Card Sort: Matching Inverses
Your teacher will give you a set of cards with numbers on them.
- Match numbers with their additive inverses.
- Next, match numbers with their multiplicative inverses.
- What do you notice about the numbers and their inverses?
Summary
To solve the equation x + 8 = \text-5, we can add the opposite of 8, or -8, to each side:
Because adding the opposite of a number is the same as subtracting that number, we can also think of it as subtracting 8 from each side.
\begin{align} x + 8 &= \text-5\\ (x+ 8) + \text-8&=(\text-5)+ \text-8\\ x&=\text-13 \end{align}
We can use the same approach for this equation:
\begin{align} \text-12 & = t +\text- \frac29\\ (\text-12)+ \frac29&=\left( t+\text-\frac29\right) + \frac29\\\text-11\frac79& = t\end{align}
To solve the equation 8x = \text-5, we can multiply each side by the reciprocal of 8, or \frac18:
Because multiplying by the reciprocal of a number is the same as dividing by that number, we can also think of it as dividing by 8.
\begin{align} 8x & = \text-5\\ \frac18 ( 8x )&= \frac18 (\text-5)\\ x&=\text-\frac58 \end{align}
We can use the same approach for this equation:
\begin{align} \text-12& =\text-\frac29 t\\ \text-\frac92\left( \text-12\right)&= \text-\frac92 \left(\text-\frac29t\right) \\ 54& = t\end{align}
Glossary Entries
- variable
A variable is a letter that represents a number. You can choose different numbers for the value of the variable.
For example, in the expression 10-x, the variable is x. If the value of x is 3, then 10-x=7, because 10-3=7. If the value of x is 6, then 10-x=4, because 10-6=4.