Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. A couple of friends decide to race each other. Leon can run 4 yards per second, whereas Michael can run 7 yards per second. Because he is slower, Leon also gets a head start of 39 yards. Shortly after they start running, Michael will catch up to Leon. How far will Michael have to run? How long will that take? Michael will catch up to Leon after running \( \square \) yards in \( \square \) seconds.

Certainly! Let's break down the problem step by step by setting up a system of equations, solving it using substitution, and then filling in the blanks with the obtained values. ### **Step 1: Define the Variables** - Let \( t \) be the time in seconds it takes for Michael to catch up to Leon. - Let \( d \) be the distance in yards that Michael runs to catch up. ### **Step 2: Set Up the Equations** 1. **Leon’s Distance:** - Leon runs at a speed of 4 yards per second. - He also has a head start of 39 yards. - Therefore, Leon's total distance can be expressed as: \[ \text{Leon’s Distance} = 4t + 39 \] 2. **Michael’s Distance:** - Michael runs at a speed of 7 yards per second. - Therefore, Michael's distance is: \[ \text{Michael’s Distance} = 7t \] ### **Step 3: Set the Equations Equal** Since Michael catches up to Leon when both have run the same distance: \[ 7t = 4t + 39 \] ### **Step 4: Solve for \( t \)** \[ 7t - 4t = 39 \\ 3t = 39 \\ t = \frac{39}{3} \\ t = 13 \text{ seconds} \] ### **Step 5: Find \( d \) (Michael’s Distance)** \[ d = 7t = 7 \times 13 = 91 \text{ yards} \] ### **Final Answer** Michael will catch up to Leon after running **91** yards in **13** seconds.