Question

1. solve the system by graphing.

$y = -x + 5$
$y = x^2 - 6x - 1$

Explanation:

Step1: Analyze the first equation

The first equation is \( y = -x + 5 \), which is a linear equation in slope - intercept form (\( y=mx + b \)) with slope \( m=- 1 \) and y - intercept \( b = 5 \). To graph this line, we can find two points. When \( x = 0 \), \( y=5 \). When \( y = 0 \), \(0=-x + 5\), so \(x = 5\). So the line passes through the points \((0,5)\) and \((5,0)\).

Step2: Analyze the second equation

The second equation is \( y=x^{2}-6x - 1 \), which is a quadratic equation. We can rewrite it in vertex form by completing the square. \(y=x^{2}-6x - 1=(x^{2}-6x + 9)-9 - 1=(x - 3)^{2}-10\). So the vertex of the parabola is at \((3,-10)\). To find other points, we can plug in some x - values. When \(x = 0\), \(y=-1\). When \(x = 6\), \(y=6^{2}-6\times6-1=-1\). When \(x = 1\), \(y=1 - 6-1=-6\). When \(x = 5\), \(y=25-30 - 1=-6\).

Step3: Find the intersection points

To solve the system by graphing, we look for the points where the line \(y=-x + 5\) and the parabola \(y=x^{2}-6x - 1\) intersect. We can also solve the system algebraically to check. Set \(-x + 5=x^{2}-6x - 1\). Rearranging gives \(x^{2}-5x - 6 = 0\). Factoring, we get \((x - 6)(x+1)=0\). So \(x=6\) or \(x=-1\).

When \(x = 6\), \(y=-6 + 5=-1\). When \(x=-1\), \(y=-(-1)+5=6\).

Answer:

The solutions of the system are \((-1,6)\) and \((6,-1)\)