Question

which of the following lines are parallel to $y = \frac{4}{3}x - 8$? select them all.
$y - 4 = -3(x + 4)$
$8x - 6y = 12$
$y = \frac{3}{4}x + 1$
$y - 1 = \frac{4}{3}(x + 6)$

Explanation:

To determine which lines are parallel to \( y = \frac{4}{3}x - 8 \), we use the fact that parallel lines have the same slope. The slope of \( y = \frac{4}{3}x - 8 \) is \( \frac{4}{3} \). We analyze each option:

Option 1: \( y - 4 = -3(x + 4) \)

• Rewrite in slope - intercept form (\( y=mx + b \)):
• Expand the right - hand side: \( y-4=-3x - 12 \)
• Add 4 to both sides: \( y=-3x - 8 \)
• The slope \( m=-3 \), which is not equal to \( \frac{4}{3} \). So this line is not parallel.

Option 2: \( 8x - 6y = 12 \)

• Solve for \( y \) to get it in slope - intercept form:
• Subtract \( 8x \) from both sides: \( - 6y=-8x + 12 \)
• Divide every term by \( - 6 \): \( y=\frac{-8x}{-6}+\frac{12}{-6} \)
• Simplify: \( y=\frac{4}{3}x-2 \)
• The slope \( m = \frac{4}{3} \), which is equal to the slope of the given line. So this line is parallel.

Option 3: \( y=\frac{3}{4}x + 1 \)

• The slope of this line is \( m=\frac{3}{4} \), which is not equal to \( \frac{4}{3} \). So this line is not parallel.

Option 4: \( y - 1=\frac{4}{3}(x + 6) \)

• Rewrite in slope - intercept form:
• Expand the right - hand side: \( y - 1=\frac{4}{3}x+8 \)
• Add 1 to both sides: \( y=\frac{4}{3}x + 9 \)
• The slope \( m=\frac{4}{3} \), which is equal to the slope of the given line. So this line is parallel.

So the lines that are parallel to \( y=\frac{4}{3}x - 8 \) are \( 8x - 6y = 12 \) and \( y - 1=\frac{4}{3}(x + 6) \).

Answer:

• \( 8x - 6y = 12 \)
• \( y - 1=\frac{4}{3}(x + 6) \)