Question

Review the graphs of a system of two linear equations in two variables: x−5y=7
x
−
5
y
=
7
and 3x−2y=−4
3
x
−
2
y
=
−
4
. Find the solution to both equations.

Answer 1

To find the solution to the system of linear equations given by:

1. \( x - 5y = 7 \)
2. \( 3x - 2y = -4 \)

we can solve these equations using either the substitution method, the elimination method, or by graphing them. Here, I'll use the substitution method for clarity.

### Step 1: Solve one equation for one variable

Let's solve the first equation for \( x \):

\[
x = 5y + 7
\]

### Step 2: Substitute into the second equation

Next, we substitute this expression for \( x \) into the second equation:

\[
3(5y + 7) - 2y = -4
\]

### Step 3: Simplify and solve for \( y \)

Expanding this gives:

\[
15y + 21 - 2y = -4
\]

Now combine like terms:

\[
13y + 21 = -4
\]

Subtracting 21 from both sides results in:

\[
13y = -4 - 21
\]
\[
13y = -25
\]

Now, divide by 13 to solve for \( y \):

\[
y = -\frac{25}{13}
\]

### Step 4: Substitute back to find \( x \)

Now that we have \( y \), we can substitute back to find \( x \):

\[
x = 5\left(-\frac{25}{13}\right) + 7
\]
\[
x = -\frac{125}{13} + \frac{91}{13} = -\frac{125 - 91}{13} = -\frac{34}{13}
\]

### Conclusion

So the solution to the system of equations is:

\[
x = -\frac{34}{13}, \quad y = -\frac{25}{13}
\]

In ordered pair notation, the solution is:

\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]