Question

solve the following system by substitution.
{y = x^2 + 11x + 111
y + 8x = 33

select the correct choice below and fill in any answer boxes in your choice.
a. the solution(s) is/are (type an ordered - pair. use a comma to separate answers as needed.)
b. there are no solutions.

Explanation:

Step1: Substitute y in the second - equation

Substitute \(y=x^{2}+11x + 111\) into \(y + 6x=33\). We get \(x^{2}+11x + 111+6x=33\).

Step2: Simplify the equation

Combine like - terms: \(x^{2}+(11x + 6x)+111 - 33=0\), which simplifies to \(x^{2}+17x + 78 = 0\).

Step3: Factor the quadratic equation

Factor \(x^{2}+17x + 78\) as \((x + 6)(x+13)=0\).

Step4: Solve for x

Set each factor equal to zero:
If \(x + 6=0\), then \(x=-6\); if \(x + 13=0\), then \(x=-13\).

Step5: Solve for y

When \(x=-6\), substitute into \(y + 6x=33\), so \(y+6\times(-6)=33\), \(y-36 = 33\), \(y=33 + 36=69\).
When \(x=-13\), substitute into \(y + 6x=33\), so \(y+6\times(-13)=33\), \(y-78 = 33\), \(y=33 + 78 = 111\).

Answer:

A. The solution(s) is/are \((-6,69),(-13,111)\)