Chapter 0.1, Problem 27E

(a)

To determine

To find the slope of the line $\frac{x}{3}+\frac{y}{4}=1$ .

Answer to Problem 27E

The slope of the line $\frac{x}{3}+\frac{y}{4}=1$ is $-\frac{4}{3}$ .

Explanation of Solution

Given the equation of a line:

  $\frac{x}{3}+\frac{y}{4}=1$

Concept Used:

Slope-Intercept form of a line:

Suppose that the equation of a line is given in the slope-intercept form: $y=mx+c$ .

Then, the $m$ gives the slope of the line and $c$ gives the $y$ -intercept of the line.

Calculation:

Given the equation of the line:

  $\frac{x}{3}+\frac{y}{4}=1$ .

Write $\frac{x}{3}+\frac{y}{4}=1$ in the slope intercept form:

  $\begin{array}{c}\frac{x}{3}+\frac{y}{4}=1\\ 12\cdot \left(\frac{x}{3}+\frac{y}{4}\right)=12\cdot 1\\ 12\cdot \frac{x}{3}+12\cdot \frac{y}{4}=12\\ 4x+3y=12\\ 3y=12-4x\\ y=\frac{12-4x}{3}\\ y=-\frac{4}{3}x+\frac{12}{3}\\ y=-\frac{4}{3}x+4\end{array}$ .

Thus, the slope-intercept form of the line:

  $y=-\frac{4}{3}x+4$ .

Compare $y=-\frac{4}{3}x+4$ with the general slope-intercept form and observe that the slope of the line is $-\frac{4}{3}$ .

Conclusion:

The slope of the line $\frac{x}{3}+\frac{y}{4}=1$ is $-\frac{4}{3}$ .

(b)

To determine

To find the $y$ -intercept of the line $\frac{x}{3}+\frac{y}{4}=1$ .

Answer to Problem 27E

The $y$ -intercept of the line $\frac{x}{3}+\frac{y}{4}=1$ is $4$ .

Explanation of Solution

Given the equation of a line:

  $\frac{x}{3}+\frac{y}{4}=1$

Concept Used:

Slope-Intercept form of a line:

Suppose that the equation of a line is given in the slope-intercept form: $y=mx+c$ .

Then, the $m$ gives the slope of the line and $c$ gives the $y$ -intercept of the line.

Calculation:

Given the equation of the line:

  $\frac{x}{3}+\frac{y}{4}=1$ .

Write $\frac{x}{3}+\frac{y}{4}=1$ in the slope intercept form:

  $\begin{array}{c}\frac{x}{3}+\frac{y}{4}=1\\ 12\cdot \left(\frac{x}{3}+\frac{y}{4}\right)=12\cdot 1\\ 12\cdot \frac{x}{3}+12\cdot \frac{y}{4}=12\\ 4x+3y=12\\ 3y=12-4x\\ y=\frac{12-4x}{3}\\ y=-\frac{4}{3}x+\frac{12}{3}\\ y=-\frac{4}{3}x+4\end{array}$ .

Thus, the slope-intercept form of the line:

  $y=-\frac{4}{3}x+4$ .

Compare $y=-\frac{4}{3}x+4$ with the general slope-intercept form and observe that the $y$ -intercept of the line is $4$ .

Conclusion:

The $y$ -intercept of the line $\frac{x}{3}+\frac{y}{4}=1$ is $4$ .

(c)

To determine

To graph the line $\frac{x}{3}+\frac{y}{4}=1$ .

Explanation of Solution

Given the line:

  $\frac{x}{3}+\frac{y}{4}=1$ .

Graph the line with the help of a graphing calculator: