Question

Which equations are equivalent to y = two-thirds x minus 4 when written in slope-intercept form? Check all that apply.
3 x minus 2 y = 4
2 x minus 3 y = 12
Negative 4 (2 x minus 3 y) = negative 4 (12)
2 (x + 6) = 3 y
2 x minus 3 y = 4

Answer 1

To determine which equations are equivalent to \(y = \frac{2}{3}x - 4\) when written in slope-intercept form \(y = mx + b\), we can start by rewriting the other equations in slope-intercept form.

1. **Equation:** \(3x - 2y = 4\)
- Rearranging for \(y\):
\[
-2y = -3x + 4 \quad \Rightarrow \quad y = \frac{3}{2}x - 2
\]
- **Not equivalent.**

2. **Equation:** \(2x - 3y = 12\)
- Rearranging for \(y\):
\[
-3y = -2x + 12 \quad \Rightarrow \quad y = \frac{2}{3}x - 4
\]
- **Equivalent.**

3. **Equation:** \(-4(2x - 3y) = -4(12)\)
- Simplifying:
\[
-8x + 12y = -48 \quad \Rightarrow \quad 12y = 8x - 48 \quad \Rightarrow \quad y = \frac{2}{3}x - 4
\]
- **Equivalent.**

4. **Equation:** \(2(x + 6) = 3y\)
- Expanding:
\[
2x + 12 = 3y \quad \Rightarrow \quad 3y = 2x + 12 \quad \Rightarrow \quad y = \frac{2}{3}x + 4
\]
- **Not equivalent.**

5. **Equation:** \(2x - 3y = 4\)
- Rearranging for \(y\):
\[
-3y = -2x + 4 \quad \Rightarrow \quad y = \frac{2}{3}x - \frac{4}{3}
\]
- **Not equivalent.**

### Summary of Equivalent Equations:
- \(2x - 3y = 12\) (Equivalent)
- \(-4(2x - 3y) = -4(12)\) (Equivalent)

The final response is:
**Check the following:**
- \(2x - 3y = 12\)
- \(-4(2x - 3y) = -4(12)\)