Question

solve the following system of equations.
y = 4x + 3
y = x² - 2
solution(s) to the equation. (if no solution, type dne)

Explanation:

Step1: Set the equations equal

Since both equal \( y \), set \( 4x + 3 = x^2 - 2 \).
\( x^2 - 4x - 5 = 0 \)

Step2: Factor the quadratic

Factor \( x^2 - 4x - 5 \).
\( (x - 5)(x + 1) = 0 \)

Step3: Solve for \( x \)

Set each factor to zero:
\( x - 5 = 0 \) gives \( x = 5 \); \( x + 1 = 0 \) gives \( x = -1 \)

Step4: Find corresponding \( y \)

For \( x = 5 \): \( y = 4(5) + 3 = 23 \)
For \( x = -1 \): \( y = 4(-1) + 3 = -1 \)

Answer:

\((5, 23)\), \((-1, -1)\)