Question

solve the system of equations.
y = x² - 18x - 46
y = -19x - 4
write the coordinates in exact form. simplify all fractions and radicals.
( , )
( , )

Explanation:

Step1: Set equations equal

Since both equal \( y \), set \( x^2 - 18x - 46 = -19x - 4 \).

Step2: Simplify to quadratic

Rearrange: \( x^2 - 18x + 19x - 46 + 4 = 0 \) → \( x^2 + x - 42 = 0 \).

Step3: Factor quadratic

\( x^2 + x - 42 = (x + 7)(x - 6) = 0 \).

Step4: Solve for \( x \)

Set factors to zero: \( x + 7 = 0 \) → \( x = -7 \); \( x - 6 = 0 \) → \( x = 6 \).

Step5: Find \( y \) for \( x = -7 \)

Substitute \( x = -7 \) into \( y = -19x - 4 \): \( y = -19(-7) - 4 = 133 - 4 = 129 \).

Step6: Find \( y \) for \( x = 6 \)

Substitute \( x = 6 \) into \( y = -19x - 4 \): \( y = -19(6) - 4 = -114 - 4 = -118 \).

Answer:

\((-7, 129)\), \((6, -118)\)