Is [tex][tex]$(-4,2)$[/tex][/tex] a solution to this system of equations?

[tex]\[
\begin{array}{l}
y=-\frac{1}{4} x+1 \\
y=\frac{1}{2} x-1
\end{array}
\][/tex]

A. Yes
B. No



Answer :

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.