Question

solve the system of equations.
y = -29x - 3
y = x² - 8x - 49
write the coordinates in exact form. simplify all fractions and radicals.
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Explanation:

Step1: Set the two equations equal to each other

Since both equations equal \( y \), we can set them equal:
\( -29x - 3 = x^2 - 8x - 49 \)

Step2: Rearrange into standard quadratic form

Move all terms to one side:
\( x^2 - 8x + 29x - 49 + 3 = 0 \)
Simplify:
\( x^2 + 21x - 46 = 0 \)

Step3: Solve the quadratic equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 21 \), \( c = -46 \).
First, calculate the discriminant \( D = b^2 - 4ac \):
\( D = 21^2 - 4(1)(-46) = 441 + 184 = 625 \)
Then, find \( x \):
\( x = \frac{-21 \pm \sqrt{625}}{2(1)} = \frac{-21 \pm 25}{2} \)

Step4: Find the two values of \( x \)

For the plus sign:
\( x = \frac{-21 + 25}{2} = \frac{4}{2} = 2 \)
For the minus sign:
\( x = \frac{-21 - 25}{2} = \frac{-46}{2} = -23 \)

Step5: Find the corresponding \( y \) values

For \( x = 2 \):
Substitute into \( y = -29x - 3 \):
\( y = -29(2) - 3 = -58 - 3 = -61 \)
For \( x = -23 \):
Substitute into \( y = -29x - 3 \):
\( y = -29(-23) - 3 = 667 - 3 = 664 \)

Answer:

\( (2, -61) \)
\( (-23, 664) \)