Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Mrs. Warren, the P.E. teacher, is pairing off students to race against each other. Colton can run $5$ yards per second, and Eduardo can run $9$ yards per second. Mrs. Warren decides to give Colton a head start of $8$ yards since he runs more slowly. Once the students start running, Eduardo will quickly catch up to Colton. How far will Eduardo have to run? How long will that take? $\newline$ Eduardo will catch up to Colton after running $\_$ yards in $\_$ seconds.

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Grade 8

Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Mrs. Warren, the P.E. teacher, is pairing off students to race against each other. Colton can run

5

yards per second, and Eduardo can run

9

yards per second. Mrs. Warren decides to give Colton a head start of

8

yards since he runs more slowly. Once the students start running, Eduardo will quickly catch up to Colton. How far will Eduardo have to run? How long will that take?

\newline

Eduardo will catch up to Colton after running

\_

yards in

\_

seconds.

Define Variables: Let's define the variables for the distances that Colton and Eduardo run as $C$ and $E$ respectively, and let $t$ represent the time in seconds after they start running. We can write two equations based on the information given: $\newline$ $1$ . Colton's distance equation: Since Colton gets an $8$ -yard head start and runs at $5$ yards per second, his distance can be represented as $C = 5t + 8$ . $\newline$ $2$ . Eduardo's distance equation: Eduardo runs at $9$ yards per second, so his distance can be represented as $E = 9t$ . $\newline$ We want to find out when Eduardo catches up to Colton, which means $C = E$ . So we can set the two equations equal to each other to solve for $t$ : $\newline$ $5t + 8 = 9t$
Equations Setup: Now we will solve for $t$ using substitution: $\newline$ $5t + 8 = 9t$ $\newline$ Subtract $5t$ from both sides to get: $\newline$ $8 = 4t$ $\newline$ Now divide both sides by $4$ to solve for $t$ : $\newline$ $t = 2$
Solve for t: Now that we have the time it takes for Eduardo to catch up to Colton, we can find out how far Eduardo will have to run. We use Eduardo's distance equation: $\newline$ $E = 9t$ $\newline$ $E = 9 \times 2$ $\newline$ $E = 18$ $\newline$ So, Eduardo will have to run $18$ yards to catch up to Colton.
Find Eduardo's Distance: Finally, we check if our solution makes sense. Colton has an $8$ -yard head start and runs at $5$ yards per second. In $2$ seconds, he will have run: $\newline$ $C = 5t + 8$ $\newline$ $C = 5 \times 2 + 8$ $\newline$ $C = 10 + 8$ $\newline$ $C = 18$ $\newline$ Both Colton and Eduardo have run $18$ yards after $2$ seconds, which confirms that Eduardo catches up to Colton after running $18$ yards in $2$ seconds.