Question

the equations of three lines are given below.
line 1: 3x - 9y=-18
line 2: y = -\frac{1}{3}x + 3
line 3: y = 3x - 7
for each pair of lines, determine whether they are parallel, perpendicular, or neither.
line 1 and line 2: parallel perpendicular neither
line 1 and line 3: parallel perpendicular neither
line 2 and line 3: parallel perpendicular neither

Explanation:

Step1: Rewrite Line 1 in slope - intercept form

Starting with $3x - 9y=-18$, solve for $y$. First, subtract $3x$ from both sides: $-9y=-3x - 18$. Then divide by $-9$: $y = \frac{1}{3}x + 2$.

Step2: Recall slope - based parallel and perpendicular rules

Two lines $y = m_1x + b_1$ and $y=m_2x + b_2$ are parallel if $m_1 = m_2$ and perpendicular if $m_1\times m_2=- 1$.

Step3: Analyze Line 1 and Line 2

The slope of Line 1 ($m_1$) is $\frac{1}{3}$, and the slope of Line 2 ($m_2$) is $-\frac{1}{3}$. Since $\frac{1}{3}
eq-\frac{1}{3}$ and $\frac{1}{3}\times(-\frac{1}{3})
eq - 1$, they are neither parallel nor perpendicular.

Step4: Analyze Line 1 and Line 3

The slope of Line 1 is $\frac{1}{3}$ and the slope of Line 3 is $3$. Since $\frac{1}{3}
eq3$ and $\frac{1}{3}\times3 = 1
eq - 1$, they are neither parallel nor perpendicular.

Step5: Analyze Line 2 and Line 3

The slope of Line 2 is $-\frac{1}{3}$ and the slope of Line 3 is $3$. Since $-\frac{1}{3}\times3=-1$, they are perpendicular.

Answer:

Line 1 and Line 2: Neither
Line 1 and Line 3: Neither
Line 2 and Line 3: Perpendicular