Chapter P.4, Problem 62E

Solution

Whether the given statement is true or false.

True

Given:

The graph of any equation of the form $ax+by=c$ , where a and b are not both zero, if always a line.

Concept Used:

• The slope-intercept form of a line is $y=mx+b$, where m is the slope of the line and b is the y -intercept of the line.
• Slope of the line passing through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by $m=\frac{y_2-y_1}{x_2-x_1}$
• The slopes of two perpendicular lines are negative reciprocal of each other.
• The slopes of two parallel lines are equal to each other.

Calculation:

If $a=0$ and $b\ne 0$ , the given equation becomes

  $\begin{array}{c}ax+by=c\\ 0\cdot x+by=c\\ by=c\\ y=\frac{c}{b}\end{array}$

It gives a horizontal line.

If $a\ne 0$ and $b=0$ , the given equation becomes

  $\begin{array}{c}ax+by=c\\ ax+0\cdot y=c\\ ax=c\\ x=\frac{c}{b}\end{array}$

It gives a vertical line.

If both a and b are not zero, it follows

  $\begin{array}{c}ax+by=c\\ by=-ax+c\\ y=-\frac{a}{b}x+\frac{c}{b}\end{array}$

It is a slant line with slope $m=-\frac{a}{b}$ and y -intercept $\frac{c}{b}$ .

Thus, the given statement is true for the given a and b .