\begin{tabular}{l} Ms. Malone is writing a quiz with 17 questions where each question is worth 3,4 , or 5 points based on difficulty. \\ She wants the number of 4 -point questions to be 2 more than the number of 5 -point questions, and she wants \\ the quiz to be worth a total of 62 points. \\ \hline 10. Write a system of equations in standard form to model this situation. \\ \begin{tabular}{l} 11. Create a matrix equation. \\ 12. Solve the matrix equation. \\ \hline 13. How many 3,4 and 5 point questions could there be? \end{tabular} \\ \hline\end{tabular}

To model this situation, we can set up a system of equations based on the given conditions. Let's denote: - \( x \) as the number of 3-point questions, - \( y \) as the number of 4-point questions, and - \( z \) as the number of 5-point questions. From the given information: 1. The total number of questions is 17, so \( x + y + z = 17 \). 2. The number of 4-point questions is 2 more than the number of 5-point questions, so \( y = z + 2 \). 3. The total points from all questions is 62, so \( 3x + 4y + 5z = 62 \). Now, we can write the system of equations in standard form as: \[ \begin{cases} x + y + z = 17 \\ y = z + 2 \\ 3x + 4y + 5z = 62 \end{cases} \] Next, we will create a matrix equation and solve it to find the values of \( x \), \( y \), and \( z \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+y+z=17\\y=z+2\\3x+4y+5z=62\end{array}\right.\) - step1: Substitute the value of \(y:\) \(\left\{ \begin{array}{l}x+z+2+z=17\\3x+4\left(z+2\right)+5z=62\end{array}\right.\) - step2: Simplify: \(\left\{ \begin{array}{l}x+2z+2=17\\3x+9z+8=62\end{array}\right.\) - step3: Solve the equation: \(\left\{ \begin{array}{l}x=15-2z\\3x+9z+8=62\end{array}\right.\) - step4: Substitute the value of \(x:\) \(3\left(15-2z\right)+9z+8=62\) - step5: Simplify: \(53+3z=62\) - step6: Move the constant to the right side: \(3z=62-53\) - step7: Subtract the numbers: \(3z=9\) - step8: Divide both sides: \(\frac{3z}{3}=\frac{9}{3}\) - step9: Divide the numbers: \(z=3\) - step10: Substitute the value of \(z:\) \(x=15-2\times 3\) - step11: Calculate: \(x=9\) - step12: Substitute the value of \(z:\) \(y=3+2\) - step13: Calculate: \(y=5\) - step14: Calculate: \(\left\{ \begin{array}{l}x=9\\y=5\\z=3\end{array}\right.\) - step15: Check the solution: \(\left\{ \begin{array}{l}x=9\\y=5\\z=3\end{array}\right.\) - step16: Rewrite: \(\left(x,y,z\right) = \left(9,5,3\right)\) The solution to the system of equations is \( x = 9 \), \( y = 5 \), and \( z = 3 \). Therefore, there could be 9 questions worth 3 points, 5 questions worth 4 points, and 3 questions worth 5 points.