Chapter 4.2, Problem 49HP

(a)

To determine

To Write:

The equation $Ax+By=C$ in slope-intercept form.

Answer to Problem 49HP

The slope-intercept form is:

  $y=-\frac{A}{B}x+\frac{C}{B}$

Explanation of Solution

Given Information:

Equation of line is $Ax+By=C$

Concept Used:

Equation of slope-intercept form of a line is given by:

  $y=mx+b\mathrm{............................}(1)$

Where

b = y-intercept

m = slope

Calculation:

  $\begin{array}{l}Ax+By=C\mathrm{................................}[Original\_equation]\\ \Rightarrow Ax+By-Ax=-Ax+C\mathrm{........}[subtracting"Ax"each\_side]\\ \Rightarrow By=-Ax+C\mathrm{.........................}[simplify]\\ \Rightarrow \frac{By}{B}=\frac{-Ax}{B}+\frac{C}{B}\mathrm{......................}[divide"B"each\_side]\\ \Rightarrow y=-\frac{A}{B}x+\frac{C}{B}\mathrm{.........................}[simplify]\end{array}$

Hence, the required equation in the slope-intercept form is :

  $y=-\frac{A}{B}x+\frac{C}{B}$

(b)

To determine

To Find :

The slope of equation $Ax+By=C$ in slope-intercept form.

Answer to Problem 49HP

The slope of line is :

  $m=-\frac{A}{B}$

Explanation of Solution

Given Information:

Equation of line is $Ax+By=C$

Concept Used:

Equation of slope-intercept form of a line is given by:

  $y=mx+b\mathrm{............................}(1)$

Where

b = y-intercept

m = slope

Calculation:

From part (a), we have the equation in slope-intercept form is :

  $y=-\frac{A}{B}x+\frac{C}{B}$

On comparing the equation with the general equation in slope-intercept form ,

  $\begin{array}{l}y=mx+b\\ m=slope\\ b=y-\mathrm{int}ercept\end{array}$

We get,

  $m=-\frac{A}{B}$

Hence, the slope of given line is $-\frac{A}{B}$

(c)

To determine

To Find :

The y-intercept of equation $Ax+By=C$ in slope-intercept form.

Answer to Problem 49HP

The y-intercept of line is :

  $\frac{C}{B}$

Explanation of Solution

Given Information:

Equation of line is $Ax+By=C$

Concept Used:

Equation of slope-intercept form of a line is given by:

  $y=mx+b\mathrm{............................}(1)$

Where

b = y-intercept

m = slope

Calculation:

From part (a), we have the equation in slope-intercept form is :

  $y=-\frac{A}{B}x+\frac{C}{B}$

On comparing the equation with the general equation in slope-intercept form ,

  $\begin{array}{l}y=mx+b\\ m=slope\\ b=y-\mathrm{int}ercept\end{array}$

We get,

  $b=y-\mathrm{int}ercept=\frac{C}{B}$

Hence, the y-intercept of given line is $\frac{C}{B}$

(d)

To determine

To Find :

Is it true for all values of A,B,C ?

Answer to Problem 49HP

No

Explanation of Solution

Given Information:

Equation of line is $Ax+By=C$

Concept Used:

Equation of slope-intercept form of a line is given by:

  $y=mx+b\mathrm{............................}(1)$

Where

b = y-intercept

m = slope

and they should be “ finite “.

Calculation:

From part (a), we have the equation in slope-intercept form is :

  $y=-\frac{A}{B}x+\frac{C}{B}$

On comparing the equation with the general equation in slope-intercept form ,

  $\begin{array}{l}y=mx+b\\ m=slope\\ b=y-\mathrm{int}ercept\end{array}$

We get,

  $\begin{array}{l}b=y-\mathrm{int}ercept=\frac{C}{B}\\ m=slope=-\frac{A}{B}\end{array}$

And they should be finite .

So, we have

  $\begin{array}{l}b\ne \infty \\ m\ne \infty \\ \Rightarrow \frac{C}{B}\ne \infty \\ \Rightarrow B\ne 0\end{array}$

Hence,

  $B\ne 0$

And for all other values of A,B,C , it is well defined.