Question

question
find all solutions of the system of equations algebraically.
write your solutions as coordinate points.
$y = 2x^2 - 9x - 20$
$3x + y = 16$
answer
two solutions
and
submit answer

Explanation:

Step1: Express y from the linear equation

From \(3x + y = 16\), we get \(y = 16 - 3x\).

Step2: Substitute y into the quadratic equation

Substitute \(y = 16 - 3x\) into \(y = 2x^2 - 9x - 20\):

$$ 16 - 3x = 2x^2 - 9x - 20 $$

Step3: Rearrange into standard quadratic form

Rearrange the equation:

$$ 2x^2 - 9x + 3x - 20 - 16 = 0 $$

$$ 2x^2 - 6x - 36 = 0 $$

Divide by 2:

$$ x^2 - 3x - 18 = 0 $$

Step4: Factor the quadratic equation

Factor \(x^2 - 3x - 18\):

$$ (x - 6)(x + 3) = 0 $$

Step5: Solve for x

Set each factor to zero:

• \(x - 6 = 0 \implies x = 6\)
• \(x + 3 = 0 \implies x = -3\)

Step6: Find corresponding y values

For \(x = 6\): \(y = 16 - 3(6) = 16 - 18 = -2\)
For \(x = -3\): \(y = 16 - 3(-3) = 16 + 9 = 25\)

Answer:

\((6, -2)\) and \((-3, 25)\)