Question

a system of equations is given.
\

$$\begin{cases} y = 0.5x - 10 \\ y = (x - 2)(x + 6) \\end{cases}$$

what is a solution to the system? round the coordinates to the nearest hundredth.

Explanation:

Step1: Set equations equal

Since \( y = 0.5x - 10 \) and \( y=(x - 2)(x + 6) \), set them equal:
\( 0.5x - 10=(x - 2)(x + 6) \)

Step2: Expand right side

Expand \( (x - 2)(x + 6) \):
\( x^2 + 6x - 2x - 12 = x^2 + 4x - 12 \)
Now equation is \( 0.5x - 10 = x^2 + 4x - 12 \)

Step3: Rearrange to quadratic

Bring all terms to one side:
\( x^2 + 4x - 12 - 0.5x + 10 = 0 \)
Simplify: \( x^2 + 3.5x - 2 = 0 \)

Step4: Use quadratic formula

Quadratic formula: \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 3.5 \), \( c = -2 \)
Calculate discriminant: \( b^2 - 4ac=(3.5)^2 - 4(1)(-2)=12.25 + 8 = 20.25 \)
Square root of discriminant: \( \sqrt{20.25}=4.5 \)
Find \( x \):
\( x=\frac{-3.5\pm4.5}{2} \)
Two solutions:
\( x_1=\frac{-3.5 + 4.5}{2}=\frac{1}{2}=0.5 \)
\( x_2=\frac{-3.5 - 4.5}{2}=\frac{-8}{2}=-4 \)

Step5: Find corresponding y

For \( x = 0.5 \):
\( y = 0.5(0.5) - 10 = 0.25 - 10=-9.75 \)
For \( x = -4 \):
\( y = 0.5(-4) - 10=-2 - 10=-12 \)

Answer:

\((0.5, -9.75)\) or \((-4, -12)\) (both are solutions; nearest hundredth already satisfied)