6 Which ordered pair is a solution to this syste \( y=x^{2}-6 x+11 \) \( y=-3 x+9 \) \( (1,0) \) \( (4,0) \)

To determine which ordered pair is a solution to the system of equations: \[ \begin{cases} y = x^2 - 6x + 11 \\ y = -3x + 9 \end{cases} \] we need to find the values of \( x \) and \( y \) that satisfy both equations simultaneously. **Step 1: Set the Equations Equal to Each Other** Since both equations are equal to \( y \), set them equal: \[ x^2 - 6x + 11 = -3x + 9 \] **Step 2: Rearrange the Equation** Move all terms to one side to form a quadratic equation: \[ x^2 - 6x + 11 + 3x - 9 = 0 \\ x^2 - 3x + 2 = 0 \] **Step 3: Solve the Quadratic Equation** Factor the quadratic: \[ x^2 - 3x + 2 = (x - 1)(x - 2) = 0 \] So, the solutions for \( x \) are: \[ x = 1 \quad \text{or} \quad x = 2 \] **Step 4: Find Corresponding \( y \) Values** Plug each \( x \) back into one of the original equations, say \( y = -3x + 9 \): 1. For \( x = 1 \): \[ y = -3(1) + 9 = 6 \] So, one solution is \( (1, 6) \). 2. For \( x = 2 \): \[ y = -3(2) + 9 = 3 \] So, the other solution is \( (2, 3) \). **Step 5: Check Given Options** The provided ordered pairs are \( (1, 0) \) and \( (4, 0) \). - **For \( (1, 0) \):** - Plug \( x = 1 \) into \( y = x^2 - 6x + 11 \): \[ y = 1 - 6 + 11 = 6 \neq 0 \] - This does not satisfy the first equation. - **For \( (4, 0) \):** - Plug \( x = 4 \) into \( y = -3x + 9 \): \[ y = -12 + 9 = -3 \neq 0 \] - This does not satisfy the second equation. **Conclusion:** Neither \( (1, 0) \) nor \( (4, 0) \) is a solution to the system. The actual solutions to the system are \( (1, 6) \) and \( (2, 3) \). **Final Answer:** Neither (1, 0) nor (4, 0) satisfies both equations, so neither is a solution to the system.