a. In a card game, each round you earn either 3 points or 5 points depending on the cards you play. After 5 rounds you have 19 points. Use \( x \) for the number of 3 point rounds and \( y \) for the number of 5 point rounds. Write a system of 2 equations that describes this situation. b. Another system is solved by the point \( (7,10) \). Explain how you can check that this solution is correct.

### a. Writing the System of Equations Let's define the variables: - \( x \) = number of 3-point rounds - \( y \) = number of 5-point rounds Given the information: 1. **Total Rounds:** You played a total of 5 rounds. 2. **Total Points:** You earned a total of 19 points. Using these, we can set up the following two equations: 1. **Total Rounds Equation:** \[ x + y = 5 \] *This equation states that the sum of 3-point rounds and 5-point rounds is equal to 5.* 2. **Total Points Equation:** \[ 3x + 5y = 19 \] *This equation represents the total points earned, where each 3-point round contributes 3 points and each 5-point round contributes 5 points.* **System of Equations:** \[ \begin{cases} x + y = 5 \\ 3x + 5y = 19 \end{cases} \] --- ### b. Verifying the Solution \((7, 10)\) To check whether \((7, 10)\) is a valid solution to the system of equations, follow these steps: 1. **Substitute \( x = 7 \) and \( y = 10 \) into the First Equation:** \[ 7 + 10 = 5 \\ 17 = 5 \quad \text{(This is FALSE)} \] 2. **Substitute \( x = 7 \) and \( y = 10 \) into the Second Equation:** \[ 3(7) + 5(10) = 19 \\ 21 + 50 = 19 \\ 71 = 19 \quad \text{(This is FALSE)} \] Since substituting \((7, 10)\) into both equations does not satisfy either equation, \((7, 10)\) is **not** a solution to the system. **Note:** It's possible that \((7, 10)\) refers to a different system of equations. To verify it correctly: - **Identify the Specific System:** Ensure you know which equations you're testing. - **Substitute the Values:** Plug \( x = 7 \) and \( y = 10 \) into each equation of that system. - **Check for Validity:** Both equations should hold true after substitution. If both equations are satisfied after substitution, then \((7, 10)\) is a valid solution. Otherwise, it is not.