Question

review the Graphs of a system of two linear equation in two variables: x-5y=7 and 3x-2y=4. Find the solution to both equations

what is the intersection points

Answer 1

To review the system of two linear equations and find the solution, we can write down the equations first:

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

### Finding the Intersection Point

The solution (or intersection point) of these equations can be found using various methods such as substitution, elimination, or graphing. Here, I will use the substitution method.

#### Step 1: Solve the first equation for \( x \)

From Equation 1:
\[
x = 5y + 7
\]

#### Step 2: Substitute \( x \) into the second equation

Now substitute \( x \) in Equation 2:
\[
3(5y + 7) - 2y = 4
\]

Expanding the equation:
\[
15y + 21 - 2y = 4
\]
\[
13y + 21 = 4
\]

#### Step 3: Solve for \( y \)

Now, isolate \( y \):
\[
13y = 4 - 21
\]
\[
13y = -17
\]
\[
y = -\frac{17}{13}
\]

#### Step 4: Substitute \( y \) back into the expression for \( x \)

Now plug \( y \) back into the expression for \( x \):
\[
x = 5\left(-\frac{17}{13}\right) + 7
\]
\[
x = -\frac{85}{13} + \frac{91}{13}
\]
\[
x = \frac{6}{13}
\]

### Intersection Point

The intersection point of the two equations, which is the solution to the system, is:
\[
\left( \frac{6}{13}, -\frac{17}{13} \right)
\]

### Summary

- The solution to the system of equations \( x - 5y = 7 \) and \( 3x - 2y = 4 \) is \( \left( \frac{6}{13}, -\frac{17}{13} \right) \). This point is where the two lines represented by the equations intersect on a graph.