Chapter 5.6, Problem 2P

a.

To determine

To determine whether the graph of the given equations is parallel/perpendicular/neither.

Answer to Problem 2P

The lines are neither parallel nor perpendicular to each other.

Explanation of Solution

Given :

The equations are

  $y=\frac{3}{4}x+7$ and $4x-3y=9$

Definition :

Two lines $y=m_{1}x+c$ and $y=m_{2}x+d$ are parallel to each other if $m_1=m_2$ .

Two lines $y=m_{1}x+c$ and $y=m_{2}x+d$ are perpedicular to each other if $m_1\times m_2=-1$ .

Calculation :

Let,

  $y=\frac{3}{4}x+7$ …………(i)

And, $4x-3y=9$ can also be written as $y=\frac{4}{3}x-3$ ………..(ii)

Comparing (i) and (ii) with the standard equation $y=mx+c$

  $m_1=\frac{3}{4}$ and $m_2=\frac{4}{3}$ are slopes of (i) and (ii) respectively.

Clearly, the slopes are neither equal to each other nor negative reciprocals of each other.

Hence, the lines are neither parallel nor perpendicular to each other.

b.

To determine

To determine whether the graph of the given equations is parallel/perpendicular/neither.

Answer to Problem 2P

Both the lines are parallel to each other.

Explanation of Solution

Given :

The equations are

  $6y=-x+6$ and $y=-\frac{1}{6}x+6$

Definition :

Two lines $y=m_{1}x+c$ and $y=m_{2}x+d$ are parallel to each other if $m_1=m_2$ .

Two lines $y=m_{1}x+c$ and $y=m_{2}x+d$ are perpedicular to each other if $m_1\times m_2=-1$ .

Calculation :

The first equation $6y=-x+6$ can also be written as

  $y=-\frac{1}{6}x+1$ …………(i)

And, $y=-\frac{1}{6}x+6$ ………(ii)

Comparing (i) and (ii) with the standard equation $y=mx+c$

  $m_1=-\frac{1}{6}$ and $m_2=-\frac{1}{6}$ are the slopes of (i) and (ii) respectively.

Clearly, the slopes of (i) and (ii) are equal to each other.

Hence, both the lines are parallel to each other.