Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = 3x^2 + 22x - 18$
$18 = x - y$
answer
two solutions
and
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Explanation:

Step1: Express y from the second equation

From \(18 = x - y\), we can rewrite it as \(y = x - 18\).

Step2: Substitute y into the first equation

Substitute \(y = x - 18\) into \(y = 3x^{2}+22x - 18\), we get:
\(x - 18 = 3x^{2}+22x - 18\)

Step3: Simplify the equation

Subtract \(x - 18\) from both sides:
\(0 = 3x^{2}+22x - 18-(x - 18)\)
\(0 = 3x^{2}+22x - 18 - x + 18\)
\(0 = 3x^{2}+21x\)

Step4: Factor the equation

Factor out \(3x\):
\(3x(x + 7)=0\)

Step5: Solve for x

Set each factor equal to zero:

• \(3x = 0\) gives \(x = 0\)
• \(x + 7 = 0\) gives \(x = - 7\)

Step6: Find the corresponding y values

For \(x = 0\), substitute into \(y = x - 18\), we get \(y = 0 - 18=-18\)
For \(x = - 7\), substitute into \(y = x - 18\), we get \(y=-7 - 18=-25\)

Answer:

\((0, - 18)\) and \((-7, - 25)\)