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

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.