Question

6x−7y=37
x+14y=−9

Answer 1

To solve the system of equations:

1. \( 6x - 7y = 37 \) (Equation 1)
2. \( x + 14y = -9 \) (Equation 2)

We can use the substitution or elimination method. Here, I will use substitution.

First, let's solve Equation 2 for \( x \):

\[
x = -9 - 14y
\]

Now, substitute this expression for \( x \) into Equation 1:

\[
6(-9 - 14y) - 7y = 37
\]

Distributing the 6:

\[
-54 - 84y - 7y = 37
\]

Combine like terms:

\[
-54 - 91y = 37
\]

Now, isolate \( y \):

\[
-91y = 37 + 54
\]
\[
-91y = 91
\]
\[
y = \frac{91}{-91} = -1
\]

Now that we have \( y = -1 \), we can substitute back into the expression for \( x \):

\[
x = -9 - 14(-1)
\]
\[
x = -9 + 14
\]
\[
x = 5
\]

Thus, the solution to the system of equations is:

\[
(x, y) = (5, -1)
\]

To verify, you can substitute \( x = 5 \) and \( y = -1 \) back into the original equations:

1. For \( 6x - 7y = 37 \):
\[
6(5) - 7(-1) = 30 + 7 = 37 \quad \text{(True)}
\]

2. For \( x + 14y = -9 \):
\[
5 + 14(-1) = 5 - 14 = -9 \quad \text{(True)}
\]

The solution satisfies both equations, confirming that the solution is correct.

Final answer:
\[
(x, y) = (5, -1)
\]