Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = 2x^2 - 23x + 50$
$3x + y = 18$
answer attempt 1 out of 2
two solutions
and

Explanation:

Step1: Solve for y in the linear equation

From \(3x + y = 18\), we get \(y = 18 - 3x\).

Step2: Substitute y into the quadratic equation

Substitute \(y = 18 - 3x\) into \(y = 2x^2 - 23x + 50\):
\(18 - 3x = 2x^2 - 23x + 50\)

Step3: Rearrange into standard quadratic form

\(2x^2 - 23x + 50 - 18 + 3x = 0\)
\(2x^2 - 20x + 32 = 0\)
Divide by 2: \(x^2 - 10x + 16 = 0\)

Step4: Factor the quadratic equation

\(x^2 - 10x + 16 = (x - 2)(x - 8) = 0\)

Step5: Solve for x

Set each factor to zero:
\(x - 2 = 0\) gives \(x = 2\)
\(x - 8 = 0\) gives \(x = 8\)

Step6: Find corresponding y values

For \(x = 2\): \(y = 18 - 3(2) = 12\)
For \(x = 8\): \(y = 18 - 3(8) = -6\)

Answer:

\((2, 12)\) and \((8, -6)\)