## Section7.2Intercepts and the Standard Form of the Equation of a Line

##### The Standard Form of the Equation of a Line.

The standard form of the equation of a line is $ax+by=c$ where $a\text{,}$ $b\text{,}$ and $c$ represent real numbers. While either $a$ or $b$ can be zero, they cannot both be zero in the same equation of a line.

To determine points on the line, we replace one of the variables in the equation with a number and solve the resultant equation for the other variable. For example, consider the line with equation $3x-4y=24\text{.}$ If we wanted to determine the point on the line that has an $x$-coordinate of $4$ we would go through the following process.

\begin{align*} 3(4)-4y\amp=24\\ 12-4y\amp=24\\ 12-4y\subtractright{12}\amp=24\subtractright{12}\\ -4y\amp=12\\ \divideunder{-4y}{-4}\amp=\divideunder{12}{-4}\\ y\amp=-3 \end{align*}

We now know that the point is $(4,-3)\text{.}$

Similarly, if we wanted to know the point on the line that has a $y$-coordinate of $15\text{,}$ we would do the following.

\begin{align*} 3x-4(15)\amp=24\\ 3x-60\amp=24\\ 3x-60\addright{60}\amp=24\addright{60}\\ 3x\amp=84\\ \divideunder{3x}{3}\amp=\divideunder{84}{3}\\ x\amp=28 \end{align*}

We can conclude that the point is $(28,15)\text{.}$

##### $x$-intercepts and $y$-intercepts.

The point on a line with an $x$-coordinate of zero (if such a point exists) is called the $y$-intercept of the line. Similarly, the point with a $y$-coordinate of zero is called the $x$-intercept of the line.

For example, the $y$-intercept of the line shown in Figure 7.2.1 is $(0,-5)\text{;}$ note that this is the point where the line intersects the $y$-axis. The $x$-intercept of the same line is $(-2,0)\text{,}$ the point where the line intersects the $x$-axis.

###### Example7.2.2.

Determine the $x$-intercept of the line with equation $-11x+5y=-110\text{.}$

Solution

Since the $x$-intercept is a point on the $x$-axis, it must have a $y$-coordinate of zero. Replacing $y$ with zero and solving for $x$ results in $x=10\text{.}$ So the $x$-intercept of the line is the point $(10,0)\text{.}$

###### Example7.2.3.

Determine the $y$-intercept of the line with equation $-11x+5y=-110\text{.}$

Solution

Since the $y$-intercept is a point on the $y$-axis, it must have an $x$-coordinate of zero. Replacing $x$ with zero and solving for $y$ results in $y=-21\text{.}$ So the $y$-intercept of the line is the point $(0,-21)\text{.}$

###### Example7.2.5.

Graph the line with equation $2x-3y=-6\text{.}$

Solution

When graphing a line by hand it's useful to have at least three points with which to align your ruler. Let's consider the line with equation $2x-3y=-6\text{.}$ The $x$-intercept and $y$-intercept of this line are, respectively, $(3,0)$ and $(0,-2)\text{.}$ In the next section we discuss the idea of the slope of the line, but we can use the basic idea right now to determine a couple of more points on the line.

The line $2x-3y=-6$ in shown in Figure 7.2.6. Note that one way to describe the movement from the $y$-intercept to the $x$-intercept is "up $2\text{,}$ right $3\text{.}$" If we continue that pattern, starting at the $x$-intercept, we land at the point $(6,2)\text{,}$ which is also on the line.

Similarly, the movement from the $x$-intercept to the $y$-intercept can be described as "down $2\text{,}$ left $3\text{,}$" and continuing that pattern takes us to the point $(-3,-4)$ which is also a point on the line.

Every line has its on step-like pattern that can be executed either left-to-right or right-to-left. This phenomenon is the basis for the concept of "slope of a line" and is explored in-depth in the next section.

### ExercisesExercises

Determine the indicated ordered pairs (points) on the lines with the given equations.

###### 1.

Complete the entries in Figure 7.2.8 for the line with equation $3x-y=6\text{.}$ Also, state the $x$ and $y$ intercepts of the line.

Solution

The $x$-intercept is $(2,0)$ and the $y$-intercept is $(0,-6)\text{.}$

###### 2.

Complete the entries in Figure 7.2.10 for the line with equation $7x+4y=-8\text{.}$ Also, state the $x$ and $y$ intercepts of the line.

Solution

The $x$-intercept is $\left(-\frac{8}{7},0\right)$ and the $y$-intercept is $(0,-2)\text{.}$