Questions: Graph both equations and find their point(s) of intersection, if any. Enter as coordinates, separating your answers with commas. x^2 - 9y = -18 -2x + 3y = 3

Graph both equations and find their point(s) of intersection, if any. Enter as coordinates, separating your answers with commas.

x^2 - 9y = -18
-2x + 3y = 3

Transcript text: Graph both equations and find their point(s) of intersection, if any. Enter as coordinates, separating your answers with commas. \[ \left\{\begin{array}{l} x^{2}-9 y=-18 \\ -2 x+3 y=3 \end{array}\right. \]

Solution

Solution Steps

Step 1: Solve the second equation for _y_

We are given the system of equations:

$x^2 - 9y = -18$

$-2x + 3y = 3$

First, solve the second equation for _y_: $-2x + 3y = 3$ $3y = 2x + 3$ $y = \frac{2}{3}x + 1$

Step 2: Substitute into the first equation

Substitute the expression for _y_ from step 1 into the first equation: $x^2 - 9(\frac{2}{3}x + 1) = -18$ $x^2 - 6x - 9 = -18$ $x^2 - 6x + 9 = 0$

Step 3: Solve for _x_

This is a perfect square trinomial: $(x-3)^2 = 0$ $x-3=0$ $x=3$

Step 4: Solve for _y_

Substitute _x=3_ back into the equation for _y_ we found in step 1: $y = \frac{2}{3}(3) + 1$ $y = 2 + 1$ $y = 3$

Final Answer:

The point of intersection is $(3, 3)$.

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