To solve the system of equations:
[tex]\[ y = 2x \][/tex]
[tex]\[ y = x^2 - 8 \][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. Here are the steps:
1. Substitute [tex]\( y = 2x \)[/tex] into [tex]\( y = x^2 - 8 \)[/tex].
Given [tex]\( y = 2x \)[/tex], replace [tex]\( y \)[/tex] in the second equation:
[tex]\[ 2x = x^2 - 8 \][/tex]
2. Rearrange the equation to standard quadratic form.
Move all terms to one side to form a quadratic equation:
[tex]\[ x^2 - 2x - 8 = 0 \][/tex]
3. Factorize the quadratic equation.
We need to find two numbers that multiply to [tex]\(-8\)[/tex] and add up to [tex]\(-2\)[/tex]. These numbers are [tex]\(-4\)[/tex] and [tex]\(2\)[/tex].
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) = 0 \][/tex]
4. Solve for [tex]\( x \)[/tex].
Set each factor equal to zero:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
5. Substitute these [tex]\( x \)[/tex] values back into [tex]\( y = 2x \)[/tex] to find the corresponding [tex]\( y \)[/tex] values.
For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 2(4) = 8 \][/tex]
This gives the point [tex]\( (4, 8) \)[/tex].
For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 2(-2) = -4 \][/tex]
This gives the point [tex]\( (-2, -4) \)[/tex].
6. Combine the results.
The solutions to the system of equations are [tex]\( (4, 8) \)[/tex] and [tex]\( (-2, -4) \)[/tex].
Given the multiple-choice options, the correct answer is:
C. [tex]\((-2, -4)\)[/tex] and [tex]\((4, 8)\)[/tex]
