Question

review the graphs of a system of two linear equations in two variables: x-5y=7 and 3x-2y=-4
solve both equations

Answer 1

To solve the system of equations given by:

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

we can use either the substitution method or the elimination method. Here, we will use the substitution method first.

### Step 1: Solve for one variable

From Equation 1, we can solve for \( x \):

\[
x = 5y + 7
\]

### Step 2: Substitute into the other equation

Next, we substitute \( x \) in Equation 2 with \( 5y + 7 \):

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

Expanding this gives:

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

Combining like terms:

\[
13y + 21 = -4
\]

### Step 3: Isolate \( y \)

Now, isolate \( y \):

\[
13y = -4 - 21
\]
\[
13y = -25
\]
\[
y = -\frac{25}{13}
\]

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

Now, substitute \( y \) back into the equation for \( x \):

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

### Final Solution

The solution to the system of equations is:

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

### Optional: Graphical Representation

To graph these equations, you would convert each equation into slope-intercept form \( y = mx + b \):

1. For \( x - 5y = 7 \):
\[
-5y = -x + 7 \quad \Rightarrow \quad y = \frac{1}{5}x - \frac{7}{5}
\]

2. For \( 3x - 2y = -4 \):
\[
-2y = -3x - 4 \quad \Rightarrow \quad y = \frac{3}{2}x + 2
\]

You would plot these lines on a graph, and the intersection point would be \(\left( -\frac{34}{13}, -\frac{25}{13} \right)\). This confirms that the solution found algebraically is the point where both lines intersect.