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

Answer 1

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

1. **Write down the equations:**
\[
y = -3x + 2 \quad \text{(1)}
\]
\[
5x + 3y = 0 \quad \text{(2)}
\]

2. **Substitute equation (1) into equation (2):**
Since equation (1) expresses \(y\) in terms of \(x\), we can substitute this expression for \(y\) into equation (2).

Substitute \(y = -3x + 2\) into (2):
\[
5x + 3(-3x + 2) = 0
\]

3. **Simplify the equation:**
\[
5x - 9x + 6 = 0
\]
Combine like terms:
\[
-4x + 6 = 0
\]

4. **Solve for \(x\):**
\[
-4x = -6
\]
Divide both sides by -4:
\[
x = \frac{6}{4} = \frac{3}{2}
\]

5. **Substitute \(x\) back to find \(y\):**
Now that we have the value of \(x\), substitute \(x = \frac{3}{2}\) back into equation (1) to find \(y\).
\[
y = -3\left(\frac{3}{2}\right) + 2
\]

6. **Calculate \(y\):**
\[
y = -\frac{9}{2} + 2
\]
Convert \(2\) to a fraction with a denominator of 2:
\[
y = -\frac{9}{2} + \frac{4}{2} = -\frac{5}{2}
\]

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