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Linear equations  (Page 2/6)

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

Graph the lines: x = − 2 size 12{x= - 2} {} , and y = 3 size 12{y=3} {} .

The graph of the line x = − 2 size 12{x= - 2} {} is a vertical line that has the x-coordinate –2 no matter what the y-coordinate is. Therefore, the graph is a vertical line passing through (–2, 0).

The graph of the line y = 3 size 12{y=3} {} , is a horizontal line that has the y-coordinate 3 regardless of what the x-coordinate is. Therefore, the graph is a horizontal line that passes through (0, 3).

The Cartesian graph on the left depicts the line x = -2. The Cartesian graph on the right depicts the line y = 3.

Most students feel that the coordinates of points must always be integers. This is not true, and in real life situations, not always possible. Do not be intimidated if your points include numbers that are fractions or decimals.

Slope of a line

Section overview

In this section, you will learn to:

  1. Find the slope of a line if two points are given.
  2. Graph the line if a point and the slope are given.
  3. Find the slope of the line that is written in the form y = mx + b size 12{y= ital "mx"+b} {} .
  4. Find the slope of the line that is written in the form Ax + By = c size 12{ ital "Ax"+ ital "By"=c} {} .

In the last section, we learned to graph a line by choosing two points on the line. A graph of a line can also be determined if one point and the "steepness" of the line is known. The precise number that refers to the steepness or inclination of a line is called the slope of the line.

From previous math courses, many of you remember slope as the "rise over run," or "the vertical change over the horizontal change" and have often seen it expressed as:

rise run size 12{ { {"rise"} over {"run"} } } {} , vertical change horizontal change size 12{ { {"vertical change"} over {"horizontal change"} } } {} , Δy Δx size 12{ { {Δy} over {Δx} } } {} etc.

We give a precise definition.

If ( x 1 size 12{x rSub { size 8{1} } } {} , y 1 size 12{y rSub { size 8{1} } } {} ) and ( x 2 size 12{x rSub { size 8{2} } } {} , y 2 size 12{y rSub { size 8{2} } } {} ) are two different points on a line, then the slope of the line is

Slope = m = y 2 − y 1 x 2 − x 1 size 12{"Slope"=m= { {y rSub { size 8{2} } - y rSub { size 8{1} } } over {x rSub { size 8{2} } - x rSub { size 8{1} } } } } {}

Find the slope of the line that passes through the points (-2, 3) and (4, -1), and graph the line.

Let ( x 1 , y 1 ) = ( − 2,3 ) size 12{ \( x rSub { size 8{1} } ,y rSub { size 8{1} } \) = \( - 2,3 \) } {} and ( x 2 , y 2 ) = ( 4, − 1 ) size 12{ \( x rSub { size 8{2} } ,y rSub { size 8{2} } \) = \( 4, - 1 \) } {} then the slope

m = − 1 − 3 4 − ( − 2 ) = − 4 6 = − 2 3 size 12{m= { { - 1 - 3} over {4 - \( - 2 \) } } = - { {4} over {6} } = - { {2} over {3} } } {}

A line passing through the points (-2,3) and (4, -1) on a Cartesian graph.

To give the reader a better understanding, both the vertical change, –4, and the horizontal change, 6, are shown in the above figure.

When two points are given, it does not matter which point is denoted as ( x 1 , y 1 ) size 12{ \( x rSub { size 8{1} } ,y rSub { size 8{1} } \) } {} and which ( x 2 , y 2 ) size 12{ \( x rSub { size 8{2} } ,y rSub { size 8{2} } \) } {} . The value for the slope will be the same. For example, if we choose ( x 1 , y 2 ) = ( 4, − 1 ) size 12{ \( x rSub { size 8{1} } ,y rSub { size 8{2} } \) = \( 4, - 1 \) } {} and ( x 2 , y 2 ) = ( − 2,3 ) size 12{ \( x rSub { size 8{2} } ,y rSub { size 8{2} } \) = \( - 2,3 \) } {} , we will get the same value for the slope as we obtained earlier. The steps involved are as follows.

m = 3 − ( − 1 ) − 2 − 4 = 4 − 6 = − 2 3 size 12{m= { {3 - \( - 1 \) } over { - 2 - 4} } = { {4} over { - 6} } = - { {2} over {3} } } {}

The student should further observe that if a line rises when going from left to right, then it has a positive slope; and if it falls going from left to right, it has a negative slope.

Find the slope of the line that passes through the points (2, 3) and (2, -1), and graph.

Let ( x 1 , y 1 ) = ( 2,3 ) size 12{ \( x rSub { size 8{1} } ,y rSub { size 8{1} } \) = \( 2,3 \) } {} and ( x 2 , y 2 ) = ( 2, − 1 ) size 12{ \( x rSub { size 8{2} } ,y rSub { size 8{2} } \) = \( 2, - 1 \) } {} then the slope

m = − 1 − 3 2 − 2 = − 4 0 = undefined size 12{m= { { - 1 - 3} over {2 - 2} } = - { {4} over {0} } ="undefined"} {}

A line passing through the points (2,9) and (2, -1) on a Cartesian graph.

The slope of a vertical line is undefined.

Graph the line that passes through the point (1, 2) and has slope − 3 4 size 12{ - { {3} over {4} } } {} .

Slope equals rise run size 12{ { {"rise"} over {"run"} } } {} . The fact that the slope is − 3 4 size 12{ { { - 3} over {4} } } {} , means that for every rise of –3 units (fall of 3 units) there is a run of 4. So if from the given point (1, 2) we go down 3 units and go right 4 units, we reach the point (5, –1). The following graph is obtained by connecting these two points.

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

Alternatively, since 3 − 4 size 12{ { {3} over { - 4} } } {} represents the same number, the line can be drawn by starting at the point (1,2) and choosing a rise of 3 units followed by a run of –4 units. So from the point (1, 2), we go up 3 units, and to the left 4, thus reaching the point (–3, 5) which is also on the same line. See figure below.

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

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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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