# Linear equations

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This chapter covers principles of linear equations. After completing this chapter students should be able to: graph a linear equation; find the slope of a line; determine an equation of a line; solve linear systems; and complete application problems using linear equations.

## Chapter overview

In this chapter, you will learn to:

1. Graph a linear equation.
2. Find the slope of a line.
3. Determine an equation of a line.
4. Solve linear systems.
5. Do application problems using linear equations.

## Graphing a linear equation

Equations whose graphs are straight lines are called linear equations . The following are some examples of linear equations:

$2x-3y=6$ , $3x=4y-7$ , $y=2x-5$ , $2y=3$ , and $x-2=0$ .

A line is completely determined by two points, therefore, to graph a linear equation, we need to find the coordinates of two points. This can be accomplished by choosing an arbitrary value for $x$ or $y$ and then solving for the other variable.

Graph the line: $y=3x+2$

We need to find the coordinates of at least two points.

We arbitrarily choose $x=-1$ , $x=0$ , and $x=1$ .

If $x=-1$ , then $y=3\left(-1\right)+2$ or $-1$ . Therefore, (–1, –1) is a point on this line.

If $x=0$ , then $y=3\left(0\right)+2$ or $y=2$ . Hence the point (0, 2).

If $x=1$ , then $y=5$ , and we get the point (1, 5). Below, the results are summarized, and the line is graphed.

 $X$ -1 0 1 $Y$ -1 2 5

Graph the line: $2x+y=4$

Again, we need to find coordinates of at least two points.

We arbitrarily choose $x=-1$ , $x=0$ and $y=2$ .

If $x=-1$ , then $2\left(-1\right)+y=4$ which results in $y=6$ . Therefore, (–1, 6) is a point on this line.

If $x=0$ , then $2\left(0\right)+y=4$ , which results in $y=4$ . Hence the point (0, 4).

If $y=2$ , then $2x+2=4$ , which yields $x=1$ , and gives the point (1, 2). The table below shows the points, and the line is graphed.

 $x$ -1 0 1 $y$ 6 4 2

The points at which a line crosses the coordinate axes are called the intercepts . When graphing a line, intercepts are preferred because they are easy to find. In order to find the x-intercept, we let $y=0$ , and to find the y-intercept, we let $x=0$ .

Find the intercepts of the line: $2x-3y=6$ , and graph.

To find the x-intercept, we let $y=0$ in our equation, and solve for $x$ .

$2x-3\left(0\right)=6$
$2x-0=6$
$2x=6$
$x=3$

Therefore, the x-intercept is 3.

Similarly by letting $x=0$ , we obtain the y-intercept which is -2.

If the x-intercept is 3, and the y-intercept is –2, then the corresponding points are (3, 0) and (0, –2), respectively.

In higher math, equations of lines are sometimes written in parametric form. For example, $x=3+2t$ , $y=1+t$ . The letter $t$ is called the parameter or the dummy variable. Parametric lines can be graphed by finding values for $x$ and $y$ by substituting numerical values for $t$ .

Graph the line given by the parametric equations: $x=3+2t$ , $y=1+t$

Let $t=0$ , 1 and 2, and then for each value of $t$ find the corresponding values for $x$ and $y$ .

The results are given in the table below.

 $t$ 0 1 2 $x$ 3 5 7 $y$ 1 2 3

## Horizontal and vertical lines

When an equation of a line has only one variable, the resulting graph is a horizontal or a vertical line.

The graph of the line $x=a$ , where $a$ is a constant, is a vertical line that passes through the point ( $a$ , 0). Every point on this line has the x-coordinate $a$ , regardless of the y-coordinate.

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