Question

solve the system of equations.
y = 23x - 30
y = x² + 36x + 10
write the coordinates in exact form. simplify all fractions and radicals.
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( , )

Explanation:

Step1: Set the equations equal

Since both equal \( y \), set \( 23x - 30 = x^2 + 36x + 10 \).

Step2: Rearrange into standard quadratic form

Subtract \( 23x - 30 \) from both sides: \( x^2 + 13x + 40 = 0 \).

Step3: Factor the quadratic

Find two numbers that multiply to \( 40 \) and add to \( 13 \): \( (x + 5)(x + 8) = 0 \).

Step4: Solve for \( x \)

Set each factor to zero: \( x + 5 = 0 \Rightarrow x = -5 \); \( x + 8 = 0 \Rightarrow x = -8 \).

Step5: Find corresponding \( y \) values

For \( x = -5 \): \( y = 23(-5) - 30 = -115 - 30 = -145 \).
For \( x = -8 \): \( y = 23(-8) - 30 = -184 - 30 = -214 \).

Answer:

\((-5, -145)\)
\((-8, -214)\)