Question

Are the pairs of lined parallel, perpendicular, or intersecting?
Y = -2/3x + 7 and -2x - 3y = 12
2x - y = 7 and 3x - 4y = -8
Y - 6 = 1-4 (x+2) and y = -4x - 10

Answer 1

To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we can convert each equation into slope-intercept form (y = mx + b), where m represents the slope of the line.

1. **For the lines \(y = -\frac{2}{3}x + 7\) and \(-2x - 3y = 12\)**:
- The first line is already in slope-intercept form. Its slope \(m_1 = -\frac{2}{3}\).
- For the second line, rearranging \(-2x - 3y = 12\):
\[
-3y = 2x + 12 \implies y = -\frac{2}{3}x - 4
\]
Its slope \(m_2 = -\frac{2}{3}\).

Since both slopes are equal (\(m_1 = m_2\)), the lines are **parallel**.

2. **For the lines \(2x - y = 7\) and \(3x - 4y = -8\)**:
- First line rearranged:
\[
-y = -2x + 7 \implies y = 2x - 7
\]
Here, the slope \(m_1 = 2\).

- Second line rearranged:
\[
3x - 4y = -8 \implies -4y = -3x - 8 \implies y = \frac{3}{4}x + 2
\]
Here, the slope \(m_2 = \frac{3}{4}\).

Since \(m_1 = 2\) and \(m_2 = \frac{3}{4}\), and since \(m_1 \cdot m_2 \neq -1\), the lines are **intersecting**.

3. **For the lines \(y - 6 = -4(x + 2)\) and \(y = -4x - 10\)**:
- First line:
\[
y - 6 = -4(x + 2) \implies y - 6 = -4x - 8 \implies y = -4x - 2
\]
Here, the slope \(m_1 = -4\).

- Second line is already in slope-intercept form, with \(m_2 = -4\).

Since both slopes are equal (\(m_1 = m_2 = -4\)), the lines are **parallel**.

### Summary:
1. \(y = -\frac{2}{3}x + 7\) and \(-2x - 3y = 12\): **Parallel**
2. \(2x - y = 7\) and \(3x - 4y = -8\): **Intersecting**
3. \(y - 6 = -4(x + 2)\) and \(y = -4x - 10\): **Parallel**