Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.

$y = -x^2 + 8x - 7$
$1 = x - y$

answer
attempt 1 out of 2
two solutions
and

Explanation:

Step1: Rearrange linear equation for $y$

From $1 = x - y$, solve for $y$:
$y = x - 1$

Step2: Substitute $y$ into quadratic equation

Replace $y$ in $y = -x^2 + 8x - 7$:
$x - 1 = -x^2 + 8x - 7$

Step3: Rearrange to standard quadratic form

Bring all terms to one side:
$x^2 - 7x + 6 = 0$

Step4: Factor the quadratic equation

Factor the polynomial:
$(x - 1)(x - 6) = 0$

Step5: Solve for $x$ values

Set each factor equal to 0:
$x - 1 = 0 \implies x=1$; $x - 6 = 0 \implies x=6$

Step6: Find corresponding $y$ values

For $x=1$: $y = 1 - 1 = 0$
For $x=6$: $y = 6 - 1 = 5$

Answer:

$(1, 0)$ and $(6, 5)$