Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = x^2 - 9x + 14$
$2 = x - y$
answer attempt 1 out of 2
two solutions
and

Explanation:

Step1: Express y from the second equation

From \(2 = x - y\), we can solve for \(y\): \(y = x - 2\)

Step2: Substitute y into the first equation

Substitute \(y = x - 2\) into \(y = x^2 - 9x + 14\):
\(x - 2 = x^2 - 9x + 14\)

Step3: Rearrange into quadratic equation

Rearrange the equation to standard quadratic form \(ax^2+bx+c = 0\):
\(x^2 - 9x + 14 - x + 2 = 0\)
\(x^2 - 10x + 16 = 0\)

Step4: Factor the quadratic equation

Factor \(x^2 - 10x + 16\):
We need two numbers that multiply to \(16\) and add to \(- 10\). The numbers are \(-2\) and \(-8\).
So, \((x - 2)(x - 8)=0\)

Step5: Solve for x

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

Step6: Find corresponding y values

For \(x = 2\), use \(y = x - 2\): \(y = 2 - 2 = 0\)
For \(x = 8\), use \(y = x - 2\): \(y = 8 - 2 = 6\)

Answer:

\((2, 0)\) and \((8, 6)\)