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In this chapter, you will learn to:
Equations whose graphs are straight lines are called linear equations . The following are some examples of linear equations:
$\mathrm{2x}-\mathrm{3y}=6$ , $\mathrm{3x}=\mathrm{4y}-7$ , $y=\mathrm{2x}-5$ , $\mathrm{2y}=3$ , 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: $y=\mathrm{3x}+2$
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 $y=3(-1)+2$ or $-1$ . Therefore, (–1, –1) is a point on this line.
If $x=0$ , then $y=3(0)+2$ 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: $\mathrm{2x}+y=4$
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 $2(-1)+y=4$ which results in $y=6$ . Therefore, (–1, 6) is a point on this line.
If $x=0$ , then $2(0)+y=4$ , which results in $y=4$ . Hence the point (0, 4).
If $y=2$ , then $\mathrm{2x}+2=4$ , 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: $\mathrm{2x}-\mathrm{3y}=6$ , and graph.
To find the x-intercept, we let $y=0$ in our equation, and solve for $x$ .
Therefore, the x-intercept is 3.
Similarly by letting $x=0$ , we obtain the y-intercept which is -2.
In higher math, equations of lines are sometimes written in parametric form. For example, $x=3+\mathrm{2t}$ , $y=1+t$ . 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: $x=3+\mathrm{2t}$ , $y=1+t$
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 |
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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