Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Mrs. Landry, the P.E. teacher, is pairing off students to race against each other. Patrick can run $4\,\text{meters per second}$ , and Camille can run $5\,\text{meters per second}$ . Mrs. Landry decides to give Patrick a head start of $26\,\text{meters}$ since he runs more slowly. Once the students start running, Camille will quickly catch up to Patrick. How long will that take? How far will Camille have to run? $\newline$ It will take $\_\_\_$ seconds for Camille to run $\_\_\_$ meters and catch up to Patrick.

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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. Landry, the P.E. teacher, is pairing off students to race against each other. Patrick can run

4\,\text{meters per second}

, and Camille can run

5\,\text{meters per second}

. Mrs. Landry decides to give Patrick a head start of

26\,\text{meters}

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

\newline

It will take

\_\_\_

seconds for Camille to run

\_\_\_

meters and catch up to Patrick.

Define Variables: Define the variables for the system of equations. $\newline$ Let $t$ represent the time in seconds after both students start running. $\newline$ Let $d$ represent the distance in meters that Camille runs. $\newline$ Patrick's head start is $26$ meters.
Patrick's Distance Equation: Write the equation for Patrick's distance. $\newline$ Patrick's speed is $4$ meters per second, and he has a $26$ -meter head start. $\newline$ The equation for Patrick's distance is: $d_P = 4t + 26$ .
Camille's Distance Equation: Write the equation for Camille's distance. $\newline$ Camille's speed is $5$ meters per second, and she does not have a head start. $\newline$ The equation for Camille's distance is: $d_C = 5t$ .
Set Up System: Set up the system of equations. $\newline$ Since Camille will catch up to Patrick, their distances will be equal at that time. $\newline$ So, we have the system of equations: $\newline$ $d_P = d_C$ $\newline$ $4t + 26 = 5t$
Solve Using Substitution: Solve the system using substitution. $\newline$ We can solve for $t$ by isolating it in one of the equations. $\newline$ Subtract $4t$ from both sides of the equation $4t + 26 = 5t$ to get: $\newline$ $26 = 5t - 4t$ $\newline$ $26 = t$
Find Camille's Distance: Find the distance Camille runs. $\newline$ Now that we know $t = 26$ , we can substitute it into Camille's distance equation to find $d_C$ . $\newline$ $d_C = 5t$ $\newline$ $d_C = 5(26)$ $\newline$ $d_C = 130$