Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = 2x^2 - 23x + 50$
$3x + y = 18$
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Explanation:

Step1: Isolate y from linear equation

Rearrange $3x + y = 18$ to solve for $y$:
$y = 18 - 3x$

Step2: Substitute y into quadratic equation

Replace $y$ in $y=2x^2-23x+50$ with $18-3x$:
$18 - 3x = 2x^2 - 23x + 50$

Step3: Simplify to standard quadratic form

Bring all terms to one side:
$2x^2 - 23x + 50 - 18 + 3x = 0$
$2x^2 - 20x + 32 = 0$
Divide by 2:
$x^2 - 10x + 16 = 0$

Step4: Factor the quadratic equation

Factor the trinomial:
$(x - 2)(x - 8) = 0$

Step5: Solve for x values

Set each factor equal to 0:
$x - 2 = 0 \implies x=2$
$x - 8 = 0 \implies x=8$

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 $(8, -6)$