Question

on the set of axes below, solve the following system of equations graphically and state the coordinates of all points in the solution set.
$y = x^2 - 12x + 29$
$2x - 2y = 14$
you can move the parabola by dragging the dots. graph the line by clicking twice.

Explanation:

Step1: Rewrite the linear equation

First, we simplify the linear equation \(2x - 2y = 14\). Divide both sides by 2: \(x - y = 7\), then solve for \(y\): \(y = x - 7\).

Step2: Find the vertex of the parabola

For the quadratic equation \(y = x^2 - 12x + 29\), the x-coordinate of the vertex is given by \(x = -\frac{b}{2a}\), where \(a = 1\) and \(b = -12\). So \(x = \frac{12}{2} = 6\). Substitute \(x = 6\) into the quadratic equation: \(y = 6^2 - 12(6) + 29 = 36 - 72 + 29 = -7\). So the vertex of the parabola is \((6, -7)\).

Step3: Find intersection points (graphically or algebraically)

To find the intersection, we can set the two equations equal: \(x^2 - 12x + 29 = x - 7\). Rearrange to get \(x^2 - 13x + 36 = 0\). Factor the quadratic: \((x - 4)(x - 9) = 0\). So \(x = 4\) or \(x = 9\).

For \(x = 4\), substitute into \(y = x - 7\): \(y = 4 - 7 = -3\).

For \(x = 9\), substitute into \(y = x - 7\): \(y = 9 - 7 = 2\).

(Alternatively, by graphing the parabola with vertex \((6, -7)\) and the line \(y = x - 7\), we can see the intersection points.)

Answer:

The solution set is \(\{(4, -3), (9, 2)\}\)