Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points. $y = x^2 - 4x - 17$ $17 = x - y$ answer two solutions and submit answer

Explanation:

Step1: Solve the second equation for y

From \(17 = x - y\), we can rearrange it to get \(y = x - 17\).

Step2: Substitute y into the first equation

Substitute \(y = x - 17\) into \(y = x^{2}-4x - 17\). So we have \(x - 17=x^{2}-4x - 17\).

Step3: Simplify the equation

Subtract \(x\) and add \(17\) to both sides: \(0=x^{2}-4x - 17 - x + 17\), which simplifies to \(x^{2}-5x = 0\).

Step4: Factor the equation

Factor out \(x\): \(x(x - 5)=0\).

Step5: Solve for x

Set each factor equal to zero: \(x = 0\) or \(x - 5=0\), so \(x = 0\) or \(x = 5\).

Step6: Find the corresponding y values

For \(x = 0\), substitute into \(y = x - 17\), we get \(y=0 - 17=-17\).
For \(x = 5\), substitute into \(y = x - 17\), we get \(y=5 - 17=-12\).

Answer:

\((0, - 17)\) and \((5, - 12)\)