Question

solve the system of equations.
$y = 10x - 45$
$y = x^2 - 2x - 10$
write the coordinates in exact form. simplify all fractions and radicals.
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Explanation:

Step1: Set the equations equal

Since both equations equal \( y \), set them equal: \( 10x - 45 = x^2 - 2x - 10 \)

Step2: Rearrange into standard quadratic

Move all terms to one side: \( x^2 - 12x + 35 = 0 \)

Step3: Factor the quadratic

Factor: \( (x - 5)(x - 7) = 0 \)

Step4: Solve for \( x \)

Set each factor to zero: \( x - 5 = 0 \) or \( x - 7 = 0 \), so \( x = 5 \) or \( x = 7 \)

Step5: Find corresponding \( y \) values

For \( x = 5 \): \( y = 10(5) - 45 = 5 \)
For \( x = 7 \): \( y = 10(7) - 45 = 25 \)

Answer:

\( (5, 5) \)
\( (7, 25) \)