Question

review the graphs of a system of two linear equations in two variables: x-5y=3 and 3x-2y= -4 find the solution to both equations

Answer 1

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

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

Let's first solve each equation for \( y \) in terms of \( x \) to facilitate graphing and understanding.

### Step 1: Solve Equation 1 for \( y \)

Starting with Equation 1:

\[
x - 5y = 3
\]

Rearranging it to isolate \( y \):

\[
-5y = 3 - x
\]

Now, divide both sides by -5:

\[
y = \frac{1}{5}x - \frac{3}{5}
\]

### Step 2: Solve Equation 2 for \( y \)

Now for Equation 2:

\[
3x - 2y = -4
\]

Rearranging to solve for \( y \):

\[
-2y = -4 - 3x
\]

Divide both sides by -2:

\[
y = \frac{3}{2}x + 2
\]

### Step 3: Find the Intersection Point

We can now set the two equations for \( y \) equal to each other to find the intersection point:

\[
\frac{1}{5}x - \frac{3}{5} = \frac{3}{2}x + 2
\]

To eliminate the fractions, multiply everything by 10:

\[
2x - 6 = 15x + 20
\]

Now simplify the equation:

\[
2x - 15x = 20 + 6
\]

\[
-13x = 26
\]

Now, solve for \( x \):

\[
x = -2
\]

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

Now substitute \( x = -2 \) back into either original equation. We can use Equation 1:

\[
-2 - 5y = 3
\]

Rearranging gives:

\[
-5y = 3 + 2
\]

\[
-5y = 5
\]

Now solve for \( y \):

\[
y = -1
\]

### Conclusion

The solution to the system of equations \( x - 5y = 3 \) and \( 3x - 2y = -4 \) is:

\[
\boxed{(-2, -1)}
\]

### Graphing

If we were to graph these two equations, we would plot the lines represented by each equation. The intersection point \((-2, -1)\) would be the point where both lines cross, representing the solution to this system of equations.