Question

which represents the solution(s) of the graphed system of equations, ( y = x^2 + x - 2 ) and ( y = 2x - 2 )?

Explanation:

Step1: Recall solution of system

The solutions of a system of equations \(y = f(x)\) and \(y = g(x)\) are the points \((x,y)\) where the graphs intersect, i.e., where \(f(x)=g(x)\) and \(y\) is the same for both.

Step2: Find intersection points

From the graph, the blue parabola \(y = x^2 + x - 2\) and the red line \(y = 2x - 2\) intersect at two points.

• First intersection: When \(x = 0\), plug into \(y = 2x - 2\), \(y=2(0)-2=-2\), so \((0, -2)\).
• Second intersection: When \(x = 1\), plug into \(y = 2x - 2\), \(y=2(1)-2 = 0\), so \((1, 0)\).

We can also solve algebraically: set \(x^2 + x - 2=2x - 2\), simplify to \(x^2 - x=0\), factor \(x(x - 1)=0\), so \(x = 0\) or \(x = 1\). Then find \(y\) for each \(x\), confirming the points \((0, -2)\) and \((1, 0)\).

Answer:

The solutions are \((0, -2)\) and \((1, 0)\)