We are given pairs of linear equations and asked to determine whether they are parallel, perpendicular, or neither. 7. $y = \frac{1}{4}x - 3$ and $y = -4x + 3$ 8. $y = 2x - 4$ and $y = -2x + 5$ 9. $3x + y = 5$ and $y = -\frac{1}{3}x + 2$ 10. $2x + 3x - 6 = 0$ and $y = -\frac{2}{3}x + 3$

1. Problem Description

We are given pairs of linear equations and asked to determine whether they are parallel, perpendicular, or neither.

7. $y = \frac{1}{4}x - 3$ and $y = -4x + 3$

8. $y = 2x - 4$ and $y = -2x + 5$

9. $3x + y = 5$ and $y = -\frac{1}{3}x + 2$

1

0. $2x + 3x - 6 = 0$ and $y = -\frac{2}{3}x + 3$

2. Solution Steps

Two lines are parallel if they have the same slope. Two lines are perpendicular if the product of their slopes is $-1$.

7. The slopes are $\frac{1}{4}$ and $-4$. Their product is $\frac{1}{4} \times -4 = -1$. Therefore, the lines are perpendicular.

8. The slopes are $2$ and $-2$. Their product is $2 \times -2 = -4 \ne -1$. The slopes are also not equal, so the lines are neither parallel nor perpendicular.

9. We need to rewrite the first equation in slope-intercept form ($y = mx + b$).

$3x + y = 5$

$y = -3x + 5$

The slopes are $-3$ and $-\frac{1}{3}$. Their product is $(-3) \times (-\frac{1}{3}) = 1 \ne -1$. The slopes are also not equal. Therefore, the lines are neither parallel nor perpendicular.

1

0. First, simplify the first equation:

$2x + 3x - 6 = 0$

$5x - 6 = 0$

$5x = 6$

$x = \frac{6}{5}$

This is a vertical line. The second line is $y = -\frac{2}{3}x + 3$, which is not a vertical line. Since the first equation is $x = \frac{6}{5}$, a vertical line, and the second equation is not a vertical line, the two lines cannot be parallel or perpendicular. Thus, they are neither.

3. Final Answer

7. Perpendicular

8. Neither

9. Neither

1

0. Neither