Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = 2x^2 - x - 153$
$x + y = 9$
answer
two solutions
and
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Explanation:

Step1: Express y from the linear equation

From \( x + y = 9 \), we get \( y = 9 - x \).

Step2: Substitute y into the quadratic equation

Substitute \( y = 9 - x \) into \( y = 2x^2 - x - 153 \):
\( 9 - x = 2x^2 - x - 153 \)

Step3: Simplify the equation

Add \( x \) to both sides: \( 9 = 2x^2 - 153 \)
Then add 153 to both sides: \( 2x^2 = 162 \)
Divide both sides by 2: \( x^2 = 81 \)

Step4: Solve for x

Take square roots: \( x = \pm 9 \)

Step5: Find corresponding y values

• When \( x = 9 \), \( y = 9 - 9 = 0 \)
• When \( x = -9 \), \( y = 9 - (-9) = 18 \)

Answer:

\((9, 0)\) and \((-9, 18)\)