Question

solve the system of equations by graphing. first graph the equations, and then fill in the solutions.
$y = -3(x - 3)^2 + 6$
$y = 3x - 3$
to graph a quadratic equation, click to plot the vertex first.
to graph a linear equation, click to plot points on the graph.
the solutions are ( , ) and ( , ).

Explanation:

Step1: Set equations equal

To find solutions, set \(-3(x - 3)^2 + 6 = 3x - 3\).

Step2: Expand the quadratic

Expand \(-3(x^2 - 6x + 9) + 6 = 3x - 3\) to \(-3x^2 + 18x - 27 + 6 = 3x - 3\).

Step3: Simplify the equation

Simplify to \(-3x^2 + 18x - 21 - 3x + 3 = 0\), then \(-3x^2 + 15x - 18 = 0\). Divide by \(-3\): \(x^2 - 5x + 6 = 0\).

Step4: Factor the quadratic

Factor \(x^2 - 5x + 6\) as \((x - 2)(x - 3) = 0\).

Step5: Solve for x

Set each factor to zero: \(x - 2 = 0\) gives \(x = 2\); \(x - 3 = 0\) gives \(x = 3\).

Step6: Find corresponding y

For \(x = 2\), \(y = 3(2) - 3 = 3\). For \(x = 3\), \(y = 3(3) - 3 = 6\).

Answer:

The solutions are \((2, 3)\) and \((3, 6)\).