Question

suppose the equations of two lines are as follows. line 1: $3x - 4y = -24$ line 2: $y = \frac{3}{4}x + 8$ a) find the slope and $y$-intercept of line 1. enter the $y$-intercept as an ordered pair. b) find the slope and $y$-intercept of line 2. enter the $y$-intercept as an ordered pair. c) are line 1 and line 2 parallel, perpendicular, or neither?

Explanation:

Part a)

Step1: Rewrite Line 1 in slope - intercept form

The equation of Line 1 is \(3x - 4y=- 24\). We want to solve for \(y\) to get it in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.
First, subtract \(3x\) from both sides: \(-4y=-3x - 24\).
Then, divide each term by \(-4\): \(y=\frac{-3x}{-4}+\frac{-24}{-4}\), which simplifies to \(y = \frac{3}{4}x+6\).

Step2: Identify slope and \(y\) - intercept

From the slope - intercept form \(y = mx + b\), for the equation \(y=\frac{3}{4}x + 6\), the slope \(m=\frac{3}{4}\) and the \(y\) - intercept \(b = 6\). The \(y\) - intercept as an ordered pair is \((0,6)\) (since when \(x = 0\), \(y=b\)).

Step1: Analyze Line 2's equation

The equation of Line 2 is \(y=\frac{3}{4}x + 8\). This is already in the slope - intercept form \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

Step2: Identify slope and \(y\) - intercept

From \(y=\frac{3}{4}x + 8\), the slope \(m = \frac{3}{4}\) and the \(y\) - intercept \(b=8\). The \(y\) - intercept as an ordered pair is \((0,8)\) (since when \(x = 0\), \(y = b\)).

Step1: Recall the conditions for parallel and perpendicular lines

• Two lines are parallel if their slopes are equal (\(m_1=m_2\)).
• Two lines are perpendicular if the product of their slopes is \(- 1\) (\(m_1\times m_2=-1\)).

Step2: Compare the slopes of Line 1 and Line 2

The slope of Line 1, \(m_1=\frac{3}{4}\), and the slope of Line 2, \(m_2=\frac{3}{4}\). Since \(m_1 = m_2=\frac{3}{4}\), the two lines satisfy the condition for parallel lines.

Answer:

Slope of Line 1: \(\frac{3}{4}\), \(y\) - intercept of Line 1: \((0,6)\)

Part b)