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A coordinate plane with four quadrants shows the x- and y-axes ranging from negative 5 to 5 in increments of 1. A solid line and a dotted line intersect each other. The equation of the solid line is x minus 5 y equals 3. The equation of the dotted line is 3 x minus 2 y equals negative 4. The intersection of both lines is shown at negative 2 on the x-axis and negative 1 on the y-axis in quadrant 3.

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

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 month ago

To find the solution to the system of equations given by \( x - 5y = 7 \) and \( 3x - 2y = -4 \), we need to solve them simultaneously.

### Step 1: Solve \( x - 5y = 7 \) for \( x \)
Rearranging the first equation:
\[
x = 5y + 7
\]

### Step 2: Substitute \( x \) into the second equation
Now, substitute \( x \) in the second equation:
\[
3(5y + 7) - 2y = -4
\]
Expanding this gives:
\[
15y + 21 - 2y = -4
\]
Simplifying:
\[
13y + 21 = -4
\]
Subtracting 21 from both sides:
\[
13y = -4 - 21
\]
\[
13y = -25
\]
Dividing by 13:
\[
y = -\frac{25}{13}
\]

### Step 3: Substitute \( y \) back into the equation for \( x \)
Now, substitute \( y \) back into \( x = 5y + 7 \):
\[
x = 5\left(-\frac{25}{13}\right) + 7
\]
Calculating \( 5\left(-\frac{25}{13}\right) \):
\[
x = -\frac{125}{13} + 7
\]
Convert 7 into a fraction to combine:
\[
7 = \frac{91}{13}
\]
Thus:
\[
x = -\frac{125}{13} + \frac{91}{13}
\]
\[
x = \frac{-125 + 91}{13}
\]
\[
x = \frac{-34}{13}
\]

### Step 4: Write the solution
The solution to the system of equations is:
\[
\left( \frac{-34}{13}, -\frac{25}{13} \right)
\]

This point represents the intersection of the two lines described by the equations \( x - 5y = 7 \) and \( 3x - 2y = -4 \).