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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineMrs. Landry, the P.E. teacher, is pairing off students to race against each other. Patrick can run 4 meters per second4\,\text{meters per second}, and Camille can run 5 meters per second5\,\text{meters per second}. Mrs. Landry decides to give Patrick a head start of 26 meters26\,\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?\newlineIt will take ___\_\_\_ seconds for Camille to run ___\_\_\_ meters and catch up to Patrick.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineMrs. Landry, the P.E. teacher, is pairing off students to race against each other. Patrick can run 4 meters per second4\,\text{meters per second}, and Camille can run 5 meters per second5\,\text{meters per second}. Mrs. Landry decides to give Patrick a head start of 26 meters26\,\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?\newlineIt will take ___\_\_\_ seconds for Camille to run ___\_\_\_ meters and catch up to Patrick.
  1. Define Variables: Define the variables for the system of equations.\newlineLet tt represent the time in seconds after both students start running.\newlineLet dd represent the distance in meters that Camille runs.\newlinePatrick's head start is 2626 meters.
  2. Patrick's Distance Equation: Write the equation for Patrick's distance.\newlinePatrick's speed is 44 meters per second, and he has a 2626-meter head start.\newlineThe equation for Patrick's distance is: dP=4t+26d_P = 4t + 26.
  3. Camille's Distance Equation: Write the equation for Camille's distance.\newlineCamille's speed is 55 meters per second, and she does not have a head start.\newlineThe equation for Camille's distance is: dC=5td_C = 5t.
  4. Set Up System: Set up the system of equations.\newlineSince Camille will catch up to Patrick, their distances will be equal at that time.\newlineSo, we have the system of equations:\newlinedP=dCd_P = d_C\newline4t+26=5t4t + 26 = 5t
  5. Solve Using Substitution: Solve the system using substitution.\newlineWe can solve for tt by isolating it in one of the equations.\newlineSubtract 4t4t from both sides of the equation 4t+26=5t4t + 26 = 5t to get:\newline26=5t−4t26 = 5t - 4t\newline26=t26 = t
  6. Find Camille's Distance: Find the distance Camille runs.\newlineNow that we know t=26t = 26, we can substitute it into Camille's distance equation to find dCd_C.\newlinedC=5td_C = 5t\newlinedC=5(26)d_C = 5(26)\newlinedC=130d_C = 130

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