Equations and Graphs of Lines
Linear equations – those equations whose graphs are lines – are prevalent in mathematics, in part because they occur frequently in our world. For example, F = 1.8C + 32 is a formula to find the Fahrenheit temperature F when the Celsius temperature C is known. This is a linear equation, which can be written in the form y = mx + b (a line with slope m and y-intercept b) or possibly, x = c, which is an equation for a vertical line.
We mentioned slope and y-intercept in the previous paragraph. What are they?
The y-intercept of a line is the point at which the line intersects with the y-axis. If you think about any point that lies on the y-axis, its x-coordinate is going to be 0. That means that if you want to find the y-intercept of a linear equation, you have only to replace x by 0, and find the y that goes with it. For example, if y = 2x – 9, and if x = 0, then y = (2)(0) – 9, or -9. Thus, for the equation y = 2x – 9, the y-intercept is -9. That means that the graph of y = 2x – 9 intersects the y-axis at the point (0, -9).
What about slope?
The slope of a line, intuitively, is a measure of its steepness. Lines that have larger slopes, either positively or negatively, are steeper than those with smaller slopes. Formally, the slope of a line can be computed if we know the coordinates of two points on that line, because two points, determine a line. Suppose we wanted to find the slope between, say, (7, 9) and (11, 49). Think about those two points, and notice that when x changes from 7 to 11, it got larger by 4 units. But the y-values from the first point to the second point change dramatically – by 40! If you imagine yourself standing on a coordinate plane at the point (7, 9), then to get to the point (11, 49), you’d need to go only 4 units to the right, and then 40 units up. This suggests a steep line because the vertical change is much more than the horizontal change. Some people call the vertical change the “rise” and the horizontal change is known as the “run.” Slope, then, is defined to be the ratio , or . To find the slope between the two points (x1, y1) and (x2, y2), we compute as . (Notice that when calculating slope, it doesn’t matter in which order you subtract the coordinates of the two points, provided you use the same order for both the top and the bottom of the fraction.)
Note that a horizontal line has a slope of 0 and a vertical line has an undefined slope. This is because for horizontal lines, the y-coordinate is always the same, so that will equal 0 for any two points on a horizontal line. On the other hand, with vertical lines, the x-coordinates are all the same. When you compute the run , you will get 0 if the line is vertical, because x2 will equal x1. That will leave the slope undefined since division by 0 is undefined.
Here’s a list of some common forms of linear equations where m denotes slope, and and are points on the line:
Notice that Point-slope and Two-point form are the same equation, with the only difference being the substitution of .
(Note: the above equations do not apply to vertical lines. A vertical line has an equation of the form )
Given a line with points (4,5) and (0,3), find its equation.
There are several different ways we can find the equation of the line. First, let’s use the Two-point form:
Now, let’s calculate the slope and use the Slope-intercept form (noting that the point (0,3) tells us that our y-intercept is 3).
So, we plug this information into the Slope-intercept form to find
This is exactly the same equation that we found using the Two-point form!
Given the equation , graph the line.
When you have to graph the line from an equation, an easy thing to do is first solve for y, and then plug in two different values for x.
Now, all we need do is choose two different values for x, say 0 and 2, and see what values y takes.
When , , and so we have the point (0,5). This point is the y-intercept of the line.
When , , which gives us the point (2,11).
Now, all we have to do is plot these two points, and then draw a straight line through them! Alternatively, observe that the graph of has y-intercept 5 and slope 3. We can graph the y-intercept at the point (0, 5) and then determine another point using the slope. If (0, 5) is on the graph, and the slope is 3, that means that for a vertical change of 3, the corresponding horizontal run will be 1. (you certainly know that 3/1 =3). So from the point (0, 5), if we move one unit to the right, we need a corresponding vertical change of 3 units up. That will land us on the point (1, 8). Connect the two points (0, 5) and (1, 8), and the line will be drawn!