Introduction to Finding Linear Equations From Graphs
The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life
—Ernest Renan (1823–1892) French Philosopher
In this lesson, you'll learn how to find the equation of a line from the graph of a line, and how to determine if an ordered pair is on a line.
Now we know how to graph the equation of a line, we will go in the other direction: Given the graph of a line, we will find its equation. Just as with input/output tables and graphing, we start with the slope. Because the slope of a line is the change in the y values between two points divided by the change in the x values of those points, we can find the slope by looking at the line and choosing any two points on it. Look at the following graph.
Although we can find the slope using any two points, choose points for which x and y are integers. On this graph, when y = 2, it is difficult to see the value of x. Its value is somewhere between 3 and 4, but we cannot be certain. However, it is easy to see that when x = 3, y = 5, and when x = 4, y = 0. We'll use these points to find the slope of the line: . The slope of the line is –5. The next step might seem familiar: To find the y-intercept, use the slope, a point on the graph, and the equation y = mx + b. Let's use the point (4,0):
0 = –5(4) + b
0 = –20 + b
20 = b
The y-intercept of the graph is 20, which means that this is the graph of the equation y = –5x + 20.
Tip:Sometimes, you can look right on the graph to find the y-intercept. If you can see where the line crosses the y-axis, you will have the y-intercept of the equation without having to perform any calculations. |
Example
To find the equation of the line graphed here, we begin again with the slope. Looking at the graph, when x = 3, y = 3, and when x = 6, y = 4. We'll use the points (3,3) and (6,4) to find the slope: . We could calculate the y-intercept, but look closely at the graph. The line crosses the y-axis where y = 2, which means that 2 is the y-intercept. This is the graph of the equation y = x + 2.
Ordered Pairs
If we are given a point, or ordered pair, and the graph of a line, we can determine if that ordered pair is on the line. We might be able to tell just by looking at the graph. If not, we can find the equation of the line, and then substitute the values of the ordered pair into the equation to see if they hold true.
Are the ordered pairs (5,38) and (–6,–46) on the following line graphed? We cannot tell just by looking, so we must find the equation of the line.
We'll use the points (0,–2) and (1,6) to find the slope: . The line crosses the y-axis where y = –2, which means that –2 is the y-intercept. This is the graph of the equation y = 8 x – 2. Now that we have the equation of the line, we will check to see if (5,38) and (–6,–46) fall on the line. Substitute 5 for x and 38 for y in the equation y = 8x – 2:
38 = 8(5) – 2 ?
38 = 40 – 2?
38 = 38
The equation holds true, so (5,38) is indeed on the line. Now, check (–6,–46):
–46 = 8(–6) – 2 ?
–46 = –48 – 2 ?
–46 ≠ –50
–46 does not equal –50, so the point (–6,–46) is not on the line y = 8x – 2.
Find practice problems and solutions for these concepts at Finding Linear Equations From Graphs Practice Questions.
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