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Section6.2Intercepts and 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 \((28,15)\text{.}\)

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 6.2.1 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.

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Figure6.2.1The \(x\) and \(y\) intercepts
Example6.2.2

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 \((10,0)\text{.}\)

Example6.2.3

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 \((0,-21)\text{.}\)

Example6.2.4

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 6.2.5. 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.

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Figure6.2.5The step-like pattern between points on a 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.

Subsection6.2.1Exercises

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

1

Complete the entries in Table 6.2.6 for the line with equation \(3x-y=6\text{.}\) Also, state the \(x\) and \(y\) intercepts of the line.

\(x\) \(y\)
\(-1\)
\(2\)
\(5\)
\(8\)
\(11\)
Table6.2.6\(3x-y=6\)
Solution

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

\(x\) \(y\)
\(-1\) \(-9\)
\(2\) \(0\)
\(5\) \(9\)
\(8\) \(18\)
\(11\) \(27\)
Table6.2.7\(3x-y=6\)
2

Complete the entries in Table 6.2.8 for the line with equation \(7x+4y=-8\text{.}\) Also, state the \(x\) and \(y\) intercepts of the line.

\(x\) \(y\)
\(2\)
\(5\)
\(-2\)
\(\frac{4}{7}\)
\(-\frac{5}{4}\)
Table6.2.8\(7x+4y=-8\)
Solution

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

\(x\) \(y\)
\(2\) \(-\frac{11}{2}\)
\(-4\) \(5\)
\(-2\) \(\frac{3}{2}\)
\(\frac{4}{7}\) \(-3\)
\(-\frac{3}{7}\) \(-\frac{5}{4}\)
Table6.2.9\(7x+4y=-8\)