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Solve the following system of equations by substitution.
$$\begin{array}{c} -x+y=-5 \\ 2 x-5 y=1 \end{array}$$
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First, we will solve the first equation for $y$.
$$\begin{array}{l} -x+y=-5 \\ y=x-5 \end{array}$$
Now, we can substitute the expression $x-5$ for $y$ in the second equation.
$$\begin{array}{l} 2 x-5 y=1 \\ 2 x-5(x-5)=1 \\ 2 x-5 x+25=1 \\ -3 x=-24 \\ x=8 \end{array}$$
Now, we substitute $x=8$ into the first equation and solve for $y$.
$$-(8)+y=-5$$
$$y=3$$
Our solution is $(8,3)$.
Check the solution by substituting $(8,3)$ into both equations.
$$\begin{array}{l} -x+y=-5 \\ -(8)+(3)=-5 \end{array}$$
True
$$2 x-5 y=1$$
$$2(8)-5(3)=1$$
True
The substitution method can be used to solve any linear system in two variables, but the method works best if one of the equa-
tions contains a coefficient of 1 or $-1$ so that we do not have to deal with fractions.
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