To determine if the point [tex]\((-4, 2)\)[/tex] is a solution to the system of equations:
[tex]\[
\begin{array}{l}
y = -\frac{1}{4}x + 1 \\
y = \frac{1}{2}x - 1
\end{array}
\][/tex]
we need to check if the point satisfies both of these equations.
Step 1: Check the first equation
Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 2\)[/tex] into the first equation:
[tex]\[
y = -\frac{1}{4}x + 1
\][/tex]
[tex]\[
2 = -\frac{1}{4}(-4) + 1
\][/tex]
Calculate the right-hand side:
[tex]\[
2 = 1 + 1
\][/tex]
[tex]\[
2 = 2
\][/tex]
The point [tex]\((-4, 2)\)[/tex] satisfies the first equation.
Step 2: Check the second equation
Now, substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 2\)[/tex] into the second equation:
[tex]\[
y = \frac{1}{2}x - 1
\][/tex]
[tex]\[
2 = \frac{1}{2}(-4) - 1
\][/tex]
Calculate the right-hand side:
[tex]\[
2 = -2 - 1
\][/tex]
[tex]\[
2 = -3
\][/tex]
The point [tex]\((-4, 2)\)[/tex] does not satisfy the second equation.
Conclusion:
The point [tex]\((-4, 2)\)[/tex] satisfies the first equation but does not satisfy the second equation. Since a solution to the system of equations must satisfy both equations simultaneously, the point [tex]\((-4, 2)\)[/tex] is not a solution to the system of equations.
Therefore, the answer is no.
