Lucid1223
Answered

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2. Do the lines given by the equations below intersect at the point [tex]\((4,-6)\)[/tex]? Explain why or why not.

[tex]\[
\begin{array}{l}
2x - y = 14 \\
x - 3y = 22
\end{array}
\][/tex]

A. No, [tex]\((4,-6)\)[/tex] is not a point of intersection because it is not a solution to [tex]\(2x - y = 14\)[/tex].

B. No, [tex]\((4,-6)\)[/tex] is not a point of intersection because it is not a solution to [tex]\(x - 3y = 22\)[/tex].

C. No, [tex]\((4,-6)\)[/tex] is not a point of intersection because it is not a solution to either equation.

D. Yes, [tex]\((4,-6)\)[/tex] is a point of intersection because it is a solution to both equations.

Sagot :

GinnyAnswer
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.
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