To solve the system of equations, we need to find the points [tex]\((x, y)\)[/tex] that satisfy both equations:
[tex]\[ y = 2x \][/tex]
[tex]\[ y = x^2 - 8 \][/tex]
We start by substituting [tex]\( y \)[/tex] from the first equation into the second equation because both are equal to [tex]\( y \)[/tex].
[tex]\[ 2x = x^2 - 8 \][/tex]
Next, we rearrange this equation to form a standard quadratic equation:
[tex]\[ x^2 - 2x - 8 = 0 \][/tex]
To solve the quadratic equation, we can factorize:
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) = 0 \][/tex]
Setting each factor to zero gives us the roots of the equation:
[tex]\[ x - 4 = 0 \implies x = 4 \][/tex]
[tex]\[ x + 2 = 0 \implies x = -2 \][/tex]
Now, we substitute each value of [tex]\( x \)[/tex] back into the first equation [tex]\( y = 2x \)[/tex] to find the corresponding [tex]\( y \)[/tex] values.
For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 2(4) = 8 \][/tex]
So one solution is [tex]\((4, 8)\)[/tex].
For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 2(-2) = -4 \][/tex]
So another solution is [tex]\((-2, -4)\)[/tex].
Thus, the solutions to the system of equations are:
[tex]\[ (-2, -4) \text{ and } (4, 8) \][/tex]
From the given choices, the correct option is:
C. [tex]\((-2, -4)\)[/tex] and [tex]\((4, 8)\)[/tex]
