Let's check whether the point (2, 2) is a solution for the given system of linear equations:
The system we are dealing with is:
[tex]\[ \begin{aligned}
& x + 3y = 8 \\
& x - 4y = -6
\end{aligned} \][/tex]
Step-by-step verification:
1. Substituting [tex]\( x = 2 \)[/tex] and [tex]\( y = 2 \)[/tex] into the first equation [tex]\( x + 3y = 8 \)[/tex]:
[tex]\[ 2 + 3(2) = 2 + 6 = 8 \][/tex]
So, for [tex]\( x + 3y \)[/tex], we get 8.
2. Substituting [tex]\( x = 2 \)[/tex] and [tex]\( y = 2 \)[/tex] into the second equation [tex]\( x - 4y = -6 \)[/tex]:
[tex]\[ 2 - 4(2) = 2 - 8 = -6 \][/tex]
So, for [tex]\( x - 4y \)[/tex], we get -6.
The step-by-step process verifies that both equations are satisfied when [tex]\( x = 2 \)[/tex] and [tex]\( y = 2 \)[/tex].
Therefore, the correct values to complete the table are:
[tex]\[
\begin{array}{cc}
x+3 y=8 & x-4 y=-6 \\
+3 \cdot (2) \stackrel{?}{=} 8 & -4 \cdot (2) \stackrel{?}{=}-6 \\
2+6 \stackrel{?}{=} 8 & 2-8 \stackrel{?}{=}-6 \\
8 \stackrel{?}{=} 8 & -6 \quad (correct)
\end{array}
\][/tex]
Thus, the apparent solution [tex]\( (2, 2) \)[/tex] is indeed correct.
