Chapter ISG, Problem 4.1.2P

a.

To determine

To calculate: The slope- intercept form of the line $h$ . And hence plot the graph and label its intercept.

Answer to Problem 4.1.2P

The slope intercept of the line $h$ is $2y=3x-5$ .

Explanation of Solution

Given information: The graph of line $l$ .

  

The equation of above line is given as:

  $Ax+By=C$ .

Here $A,B\&C$ are non -zero real numbers.

The given information of the line of which to find slope intercept form is given as:

  $\text{Line}h:A=6,C=10,\text{and}h\parallel l$ .

Calculation:

The slope-intercept form of a line $l$ is given as:

  $y=mx+b$

Here $m$ is the slope and $b$ is the $y$ -intercept.

Now the slope of given line $l$ is given as:

  $m=-\frac{A}{B}$

And from the graph, $y$ -intercept of line $l$ is given as:

  $b=-1$

So the slope-intercept form of a line $l$ is:

  $\begin{array}{c}y=\left(\frac{-A}{B}\right)x-1\left(1\right)\\ By=-Ax-B\end{array}$

As the given line $l$ passes through the point $\left(2,2\right)$ so substituting the point in the above equation become as:

  $\begin{array}{c}2B=-2A-B\\ 3B=-2A\\ \frac{-A}{B}=\frac{3}{2}\end{array}$

So the slope-intercept form of a line $l$ is given as:

  $y=\frac{3}{2}x-1$ .

By using the given information, $\text{Line}h:A=6,C=10,\text{and}h\parallel l$ the equation of line $h$ is given as:

  $6x+By=10\left(2\right)$

Slope of the line $h$ is given as:

  $m_1=\frac{-6}{B}$

Now as given $h\parallel l$ , So

Slope of both the lines are same but different $y$ -intercept.

So slope of the line $h$ is given as:

  $\begin{array}{c}{m}_1=\frac{3}{2}\\ \frac{-6}{B}=\frac{3}{2}\\ B=-4\end{array}$

And the $y$ -intercept of the line $h$ is given as:

  $\begin{array}{c}\frac{C}{B}=\frac{10}{-4}\\ \frac{C}{B}=\frac{-5}{2}\end{array}$

The slope-intercept form of a line $h$ is given as:

  $\begin{array}{l}y=\frac{3}{2}x-\frac{5}{2}\\ 2y=3x-5\end{array}$

The graph of the equation $2y=3x-5$ is given as:

  

b.

To determine

To calculate: The slope- intercept form of the line $j$ . And hence plot the graph and label its intercept.

Answer to Problem 4.1.2P

The number of ways that mathematics books must be together and the novels must be together is 72.

Explanation of Solution

Given information: The graph of line $l$ .

  

The equation of above line is given as:

  $Ax+By=C$ .

Here $A,B\&C$ are non -zero real numbers.

The given information of the line of which to find slope intercept form is given as:

  $\text{Line}j:B=1,j\perp l,\text{and has the same}y\text{intercept as}l$ .

Calculation:

The slope-intercept form of a line $l$ is given as:

  $y=mx+b$

Here $m$ is the slope and $b$ is the $y$ -intercept.

Now the slope of given line $l$ is given as:

  $m=-\frac{A}{B}$

And from the graph, $y$ -intercept of line $l$ is given as:

  $b=-1$

So the slope-intercept form of a line $l$ is:

  $\begin{array}{c}y=\left(\frac{-A}{B}\right)x-1\left(1\right)\\ By=-Ax-B\end{array}$

As the given line $l$ passes through the point $\left(2,2\right)$ so substituting the point in the above equation become as:

  $\begin{array}{c}2B=-2A-B\\ 3B=-2A\\ \frac{-A}{B}=\frac{3}{2}\end{array}$

So the slope-intercept form of a line $l$ is given as:

  $y=\frac{3}{2}x-1$ .

By using the given information, $\text{Line}j:B=1,j\perp l,\text{and has the same}y\text{intercept as}l$ the equation of line $j$ is given as:

  $Ax+1y=C\left(2\right)$

Slope of the line $h$ is given as:

  $\begin{array}{c}{m}_2=-\frac{A}{1}\\ m_2=-A\end{array}$

Now as given $j\perp l$ , So

Product of the slope of both the lines is $-1$ .

So slope of the line $j$ is given as:

  $\begin{array}{c}m{m}_2=-1\\ \frac{3}{2}\cdot \left(-A\right)=-1\\ A=\frac{2}{3}\end{array}$

As , the slope of line $j$ is $m_2=-\frac{2}{3}$ and $y$ intercept is $-1$ .

The slope-intercept form of a line $j$ is given as:

  $\begin{array}{c}y=-\frac{2}{3}x-1\\ 3y=-2x-3\end{array}$

The graph of the equation $3y=-2x-3$ is given as: