To determine if the point [tex]\((4, -6)\)[/tex] is a point of intersection for the lines given by the equations
[tex]\[ 2x - y = 14 \][/tex]
and
[tex]\[ x - 3y = 22 \][/tex]
we need to verify if [tex]\((4, -6)\)[/tex] satisfies both equations.
### Step-by-Step Verification:
1. Checking the first equation [tex]\(2x - y = 14\)[/tex]:
- Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = -6\)[/tex] into the equation.
[tex]\[
2(4) - (-6) = 14
\][/tex]
- Simplify the expression:
[tex]\[
8 + 6 = 14
\][/tex]
- The left-hand side is [tex]\(14\)[/tex], which matches the right-hand side. So, the point [tex]\((4, -6)\)[/tex] satisfies the first equation.
2. Checking the second equation [tex]\(x - 3y = 22\)[/tex]:
- Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = -6\)[/tex] into the equation.
[tex]\[
4 - 3(-6) = 22
\][/tex]
- Simplify the expression:
[tex]\[
4 + 18 = 22
\][/tex]
- The left-hand side is [tex]\(22\)[/tex], which matches the right-hand side. So, the point [tex]\((4, -6)\)[/tex] satisfies the second equation.
Since the point [tex]\((4, -6)\)[/tex] satisfies both equations, it means that the lines intersect at [tex]\((4, -6)\)[/tex].
### Conclusion:
Yes, [tex]\((4, -6)\)[/tex] is a point of intersection because it is a solution to both equations.
