Solve Systems of Equations by Elimination: example 5.32

The sum of two numbers is 39. Their difference is 9. Find the numbers.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The sum of two numbers is 42. Their difference is 8. Find the numbers.

How did it go?

The sum of two numbers is 39. Their difference is 9. Find the numbers.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The sum of two numbers is 42. Their difference is 8. Find the numbers.

How did it go?

Step 1. Read the problem.
Step 2. Identify what we are looking for.	We are looking for two numbers.
Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let $n =$ the first number. $m =$ the second number.
Step 4. Translate into a system of equations. The system is:	The sum of two numbers is 39. $n + m = 39$ Their difference is 9. $\begin{matrix} n - m = 9 \\ {\begin{matrix} n + m = 39 \\ n - m = 9 \end{matrix} \end{matrix}$
Step 5. Solve the system of equations. To solve the system of equations, use elimination. The equations are in standard form and the coefficients of $m$ are opposites. Add. Solve for $n$ . Substitute $n = 24$ into one of the original equations and solve for $m$ .	$\begin{matrix} \underset{____________}{{\begin{matrix} n + m = 39 \\ n - m = 9 \end{matrix}} \\ 2 n = 48 \\ n = 24 \\ n + m = 39 \\ 24 + m = 39 \\ m = 15 \end{matrix}$
Step 6. Check the answer.	Since $24 + 15 = 39$ and $24 - 15 = 9$ , the answers check.
Step 7. Answer the question.	The numbers are 24 and 15.

Step 1. Read the problem.
Step 2. Identify what we are looking for.	We are looking for two numbers.
Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let $n =$ the first number. $m =$ the second number.
Step 4. Translate into a system of equations. The system is:	The sum of two numbers is 39. $n + m = 39$ Their difference is 9. $\begin{matrix} n - m = 9 \\ {\begin{matrix} n + m = 39 \\ n - m = 9 \end{matrix} \end{matrix}$
Step 5. Solve the system of equations. To solve the system of equations, use elimination. The equations are in standard form and the coefficients of $m$ are opposites. Add. Solve for $n$ . Substitute $n = 24$ into one of the original equations and solve for $m$ .	$\begin{matrix} \underset{____________}{{\begin{matrix} n + m = 39 \\ n - m = 9 \end{matrix}} \\ 2 n = 48 \\ n = 24 \\ n + m = 39 \\ 24 + m = 39 \\ m = 15 \end{matrix}$
Step 6. Check the answer.	Since $24 + 15 = 39$ and $24 - 15 = 9$ , the answers check.
Step 7. Answer the question.	The numbers are 24 and 15.