Question

A system of linear equations is shown.
y = - 3x + 2
5x + 3y = 0

What is the y-coordinate of the solution to this system?

step by step, how did you get the simplifying part

Answer 1

To find the y-coordinate of the solution to the system of linear equations given by:

1. \( y = -3x + 2 \)
2. \( 5x + 3y = 0 \)

we can follow these steps:

### Step 1: Substitute for \( y \)

Since the first equation gives us \( y \) in terms of \( x \), we can substitute \( y \) from the first equation into the second equation.

We have:
\[
y = -3x + 2
\]

Now substitute this expression for \( y \) into the second equation:
\[
5x + 3(-3x + 2) = 0
\]

### Step 2: Simplify the equation

Now, let's simplify the equation. Distributing \( 3 \) into \( (-3x + 2) \):
\[
5x - 9x + 6 = 0
\]
This simplifies to:
\[
(5x - 9x) + 6 = 0
\]
\[
-4x + 6 = 0
\]

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

Next, we'll isolate \( x \):
\[
-4x = -6
\]
\[
x = \frac{-6}{-4} = \frac{3}{2}
\]

### Step 4: Find the corresponding y-coordinate

Now that we have \( x = \frac{3}{2} \), we can find the corresponding \( y \)-coordinate using the first equation:
\[
y = -3x + 2
\]
Substituting \( x = \frac{3}{2} \):
\[
y = -3\left(\frac{3}{2}\right) + 2
\]
Calculating that gives:
\[
y = -\frac{9}{2} + 2
\]
To add these, convert \( 2 \) into a fraction:
\[
y = -\frac{9}{2} + \frac{4}{2} = -\frac{9}{2} + \frac{4}{2} = -\frac{5}{2}
\]

### Conclusion

Thus, the y-coordinate of the solution to the system of equations is:
\[
\boxed{-\frac{5}{2}}
\]