Answer
$(-3,4), (4,-3)$
Work Step by Step
We are given the system:
$\begin{cases}
y=1-x\\
x^2+y^2=25
\end{cases}$
We will use the substitution method. Equation 1 is solved for $y$. Substitute the expression of $y$ in Equation 2 to eliminate $y$ and determine $x$:
$\begin{cases}
y=1-x\\
x^2+(1-x)^2=25
\end{cases}$
$x^2+(1-x)^2=25$
$x^2+1-2x+x^2=25$
$2x^2-2x+1-25=0$
$2x^2-2x-24=0$
$2(x^2-x-12)=0$
$x^2-x-12=0$
$x^2-4x+3x-12=0$
$x(x-4)+3(x-4)=0$
$(x-4)(x+3)=0$
$x-4=0\Rightarrow x_1=4$
$x+3=0\Rightarrow x_2=-3$
Substitute each of the values of $x$ in the expression of $y$ to determine $y$:
$y=1-x$
$x_1=4\Rightarrow y_1=1-4=-3$
$x_2=-3\Rightarrow x_2=1-(-3)=4$
The system's solutions are:
$(-3,4), (4,-3)$
