There are several forms that the equation of a line can take. They may look different, but they all describe the same line--a line can be described by many equations. All (linear) equations describing a particular line, however, are equivalent.
The first of the forms for a linear equation is slope-intercept form. Equations in slope-intercept form look like this:
where m is the slope of the line and b is the y-intercept of the line, or the y-coordinate of the point at which the line crosses the y-axis.
To write an equation in slope-intercept form, given a graph of that equation, pick two points on the line and use them to find the slope. This is the value of m in the equation. Next, find the coordinates of the y-intercept--this should be of the form (0, b). The y- coordinate is the value of b in the equation.
Finally, write the equation, substituting numerical values in for m and b. Check your equation by picking a point on the line (not the y-intercept) and plugging it in to see if it satisfies the equation.
Example 1: Write an equation of the following line in slope-intercept form:
First, pick two points on the line--for example, (2, 1) and (4, 0). Use these points to calculate the slope: m = =
= - .
Next, find the y-intercept: (0, 2). Thus, b = 2.
Therefore, the equation for this line is y = - x + 2.
Check
using the point (4, 0): 0 = - (4) + 2 ? Yes.
Example 2: Write an equation of the line with slope m = which crosses the y-axis at (0, -
).
y = x -
Example 3: Write an equation of the line with y-intercept 3 that is parallel to the line y = 7x -
9.
Since y = 7x - 9 is in slope-intercept form, its slope is 7.
Since parallel lines have the same slope, the slope of the new line will also be 7. m = 7. b = 3.
Thus, the equation of the line is y = 7x + 3.
Example 4: Write an equation of the line with y-intercept 4 that is perpendicular to the line 3y - x = 9.
The slope of 3y - x = 9 is .
Since the slopes of perpendicular lines are opposite reciprocals, m = - 3. b = 4.
Thus, the equation of the line is y = - 3x + 4.
An equation in the slope-intercept form is written as
$$y=mx+b$$
Where m is the slope of the line and b is the y-intercept. You can use this equation to write an equation if you know the slope and the y-intercept.
Example
Find the equation of the line
Choose two points that are on the line
Calculate the slope between the two points
$$m=\frac{y_{2}\, -y_{1}}{x_{2}\, -x_{1}}=\frac{\left (-1 \right )-3}{3-\left ( -3 \right )}=\frac{-4}{6}=\frac{-2}{3}$$
We can find the b-value, the y-intercept, by looking at the graph
b = 1
We've got a value for m and a value for b. This gives us the linear function
$$y=-\frac{2}{3}x+1$$
In many cases the value of b is not as easily read. In those cases, or if you're uncertain whether the line actually crosses the y-axis in this particular point you can calculate b by solving the equation for b and then substituting x and y with one of your two points.
We can use the example above to illustrate this. We've got the two points (-3, 3) and (3, -1). From these two points we calculated the slope
$$m=-\frac{2}{3}$$
This gives us the equation
$$y=-\frac{2}{3}x+b$$
From this we can solve the equation for b
$$b=y+\frac{2}{3}x$$
And if we put in the values from our first point (-3, 3) we get
$$b=3+\frac{2}{3}\cdot \left ( -3 \right )=3+\left ( -2 \right )=1$$
If we put in this value for b in the equation we get
$$y=-\frac{2}{3}x+1$$
which is the same equation as we got when we read the y-intercept from the graph.
To summarize how to write a linear equation using the slope-interception form you
- Identify the slope, m. This can be done by calculating the slope between two known points of the line using the slope formula.
- Find the y-intercept. This can be done by substituting the slope and the coordinates of a point (x, y) on the line in the slope-intercept formula and then solve for b.
Once you've got both m and b you can just put them in the equation at their respective position.
Video lesson
Find the equation to the graph