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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 size 12{2x - 3y=6} {} , 3x = 4y 7 size 12{3x=4y - 7} {} , y = 2x 5 size 12{y=2x - 5} {} , 2y = 3 size 12{2y=3} {} , and x 2 = 0 size 12{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 size 12{x} {} or y size 12{y} {} and then solving for the other variable.

Graph the line: y = 3x + 2 size 12{y=3x+2} {}

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

We arbitrarily choose x = 1 size 12{x= - 1} {} , x = 0 size 12{x=0} {} , and x = 1 size 12{x=1} {} .

If x = 1 size 12{x= - 1} {} , then y = 3 ( 1 ) + 2 size 12{y=3 \( - 1 \) +2} {} or 1 size 12{ - 1} {} . Therefore, (–1, –1) is a point on this line.

If x = 0 size 12{x=0} {} , then y = 3 ( 0 ) + 2 size 12{y=3 \( 0 \) +2} {} or y = 2 size 12{y=2} {} . Hence the point (0, 2).

If x = 1 size 12{x=1} {} , then y = 5 size 12{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

A line passing through the points (-1,1), (0,2) and (1,5) on a Cartesian graph.

Graph the line: 2x + y = 4 size 12{2x+y=4} {}

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

We arbitrarily choose x = 1 size 12{x= - 1} {} , x = 0 size 12{x=0} {} and y = 2 size 12{y=2} {} .

If x = 1 size 12{x= - 1} {} , then 2 ( 1 ) + y = 4 size 12{2 \( - 1 \) +y=4} {} which results in y = 6 size 12{y=6} {} . Therefore, (–1, 6) is a point on this line.

If x = 0 size 12{x=0} {} , then 2 ( 0 ) + y = 4 size 12{2 \( 0 \) +y=4} {} , which results in y = 4 size 12{y=4} {} . Hence the point (0, 4).

If y = 2 size 12{y=2} {} , then 2x + 2 = 4 size 12{2x+2=4} {} , which yields x = 1 size 12{x=1} {} , and gives the point (1, 2). The table below shows the points, and the line is graphed.

x size 12{x} {} -1 0 1
y size 12{y} {} 6 4 2
A line passing through the points (-1,6), (0,4) and (1,2) on a Cartesian graph.

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 size 12{y=0} {} , and to find the y-intercept, we let x = 0 size 12{x=0} {} .

Find the intercepts of the line: 2x 3y = 6 size 12{2x - 3y=6} {} , and graph.

To find the x-intercept, we let y = 0 size 12{y=0} {} in our equation, and solve for x size 12{x} {} .

2x 3 ( 0 ) = 6 size 12{2x - 3 \( 0 \) =6} {}
2x 0 = 6 size 12{2x - 0=6} {}
2x = 6 size 12{2x=6} {}
x = 3 size 12{x=3} {}

Therefore, the x-intercept is 3.

Similarly by letting x = 0 size 12{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.

A line passing through the points (0,-2) and (3,0) on a Cartesian graph.

In higher math, equations of lines are sometimes written in parametric form. For example, x = 3 + 2t size 12{x=3+2t} {} , y = 1 + t size 12{y=1+t} {} . The letter t size 12{t} {} is called the parameter or the dummy variable. Parametric lines can be graphed by finding values for x size 12{x} {} and y size 12{y} {} by substituting numerical values for t size 12{t} {} .

Graph the line given by the parametric equations: x = 3 + 2t size 12{x=3+2t} {} , y = 1 + t size 12{y=1+t} {}

Let t = 0 size 12{t=0} {} , 1 and 2, and then for each value of t size 12{t} {} find the corresponding values for x size 12{x} {} and y size 12{y} {} .

The results are given in the table below.

t size 12{t} {} 0 1 2
x size 12{x} {} 3 5 7
y size 12{y} {} 1 2 3

A line passing through the points (3,1), (5,2) and (7,3) on a Cartesian graph.

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 size 12{x=a} {} , where a size 12{a} {} is a constant, is a vertical line that passes through the point ( a size 12{a} {} , 0). Every point on this line has the x-coordinate a size 12{a} {} , regardless of the y-coordinate.

Questions & Answers

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Source:  OpenStax, Linear equations. OpenStax CNX. Jun 15, 2015 Download for free at https://legacy.cnx.org/content/col11828/1.1
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