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How To Find the Equation of a Line

How to Find the Equation of a Line
Co-authored by Grace Imson, MA
Updated: December 19, 2019
Finding the equation for a line is a common problem in geometry and trigonometry. There are two common situations where you are asked to find the equation for a line: either you’ll be provided with one point on the line and the slope of the line, or you’ll be provided two points on the line. In either case, finding the equation for that line isn’t difficult, provided you use the correct formula and work carefully.

Method 1 of 2:
Calculating the Equation with One Point and the Slope

1
Plug the slope in for m in the formula y-y1 = m(x-x1). This is known as the point-slope formula. The point-slope formula uses the slope and the coordinates of a point along the line to find the y-intercept. Use the slope in place of m in y-y1 = m(x-x1).[1]
For example, if you know the slope of the line is 2, then your formula will look like this: y-y1 = 2(x-x1).
EXPERT TIP

GRACE IMSON, MA
Math Instructor, City College of San Francisco
Our Expert Agrees: When you’re given two points to solve for the equation of the line, the first thing you have to find is the slope of the line. To get that, subtract the vertical coordinates, then divide that by the difference in the horizontal coordinates.

2
Replace x1 and y1 with the coordinates of the point. Use the coordinates you’re given as (x1, y1). Put the numbers in the corresponding spot on your formula before you start solving the equation.[2]
For example, if you know the coordinate is (4, 3), your formula will read: y-3 = 2(x-4).

3
Solve the formula for y to get the final slope-intercept formula. Follow the mathematical order of operations and the distributive property to remove the x-term from parenthesis.
In our example, first you’d use the distributive property to get y-3=2x-8.
Then, add 3 to each side so y is by itself.
The final equation for a line in slope-intercept form with a slope of 2 that contains the point (4, 3) is y = 2x-5.
Method 2 of 2:
Finding the Equation Using Two Points

1
Find the slope using m = (y2-y1)/(x2-x1). The ordered pairs of the coordinates you have are listed as (x, y). Use the first set of coordinates as (x1, y1) and the second set as (x2, y2). Plug the numbers into the formula m = (y2-y1)/(x2-x1) and solve for m.[3]
For example, if your coordinates are (3, 8) and (7, 12), the formula would read: m = (12-8)/(7-3) = 4/4 = 1. In this case, your slope, or m, equals 1.

2
Replace the m in the slope-intercept formula with the slope you found. The slope-intercept formula of a line is written as y = mx+b, where m is the slope and b is the y-intercept (the point on the y-axis where the line crosses it). Plug the number you found for your slope in place of m.[4]
In our example, the formula would read y = 1x+b or y = x+b when you replace the slope value.
3
Substitute x and y for one of the points you know to solve for the y-intercept. Pick one of the ordered pairs to put into the slope-intercept formula. Put the x-value in place of x and the y-value in place of y.[5]
In this example, if you chose (3, 8) as your coordinates, then the formula would read 8 = 1(3)+b.
4
Solve the equation for b. Once you plug the x- and y-values as well as your slope into the formula, find the value of b in the equation. Follow the order of operations first before moving the rest of the numbers to the other side. Leave b on one side of the equation to solve it.[6]
In our example, the formula currently reads 8 = 1(3)+b. Multiply 1 and 3 together to get 8 = 3+b. Since 3 is a positive number, subtract 3 from each side to isolate b. This leaves you with 5 = b, or b = 5.
5
Plug in the slope and y-intercept into the slope-intercept formula to finish the equation. Once you’re finished, plug in the slope for m and the y-intercept for b. After that, you’ve found the equation for the line.
For example, the equation for the line with points on (3, 8) and (7, 12) is y = 1x+5 or simply y = x+5.
Community Q&A
Question
How do I find an intercept of a line?
Donagan
Top Answerer
Using the equation, set x equal to zero and solve for y to find the y-intercept, or set y equal to zero and solve for x to find the x-intercept.
Question
What should I do with a negative y?
Community Answer
You multiply the whole equation by -1 to remove the negative sign. For example, if the question is: -y=-5x+1, you would then change the question to: -y*(-1) = (-5x+1)*(-1), and then get: y=5x-1.
Question
What about given a slope and an intercept in y?
Donagan
Top Answerer
Express the equation in standard form, y = mx + b, where m is the slope, and b is the y-intercept.
Question
The teacher wants me to write the equation of a line. If the gradient is 3 and the y intercept is -5, is the equation y = 3x + -5?
Donagan
Top Answerer
Yes, y = 3x – 5.
Question
What if my parallel line is the y-axis?
Donagan
Top Answerer
That means the line you’re looking for would be defined as x = b: the line is vertical, m is “undefined” (infinity), and b is the x-intercept.
Question
How do I find whether a point (coordinate) falls on a line?
Donagan
Top Answerer
Take the x- and y-coordinates of the point, and insert them into the equation of the line. If the equation holds true with those x and y values inserted, the point is on the line.
Question
How do I find a parallel line?
Donagan
Top Answerer
Assuming a given straight line in the form of mx + b, any parallel line would have the same form with the same “m” but a different “b.”
Question
How would I solve a system with only one given point?
Donagan
Top Answerer
A “system” consists of at least two different equations (and thus at least two lines on a graph). As the above article explains, two pieces of information are required in order to define a line: a point and a slope. To “solve a system with only one given point,” the point would have to be the intersection of the lines, and you would then need each line’s slope.
Question
What about one point and no slope?
Donagan
Top Answerer
As the above article explains, two pieces of information are required in order to define a straight line: the line’s slope and a point on the line. If you don’t know the slope, you could have an infinite number of lines passing through a given point.
Question
How would I find a line’s equation if I know one point on the line and have the equation for another parallel line?
Community Answer
You would take the slope from that line and use the above example to use the given point and the slope to figure out which point would be the slope the line would go through.
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About This Article
Grace Imson, MA
Math Instructor, City College of San Francisco
This article was co-authored by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University.
Co-authors: 58
Updated: December 19, 2019
Views: 715,241
Article Rating: 91% – 21 votes
Categories: Featured Articles | Coordinate Geometry
Article Summary
References
↑http://www.coolmath.com/algebra/08-lines/11-finding-equation-line-point-slope-01
↑http://www.coolmath.com/algebra/08-lines/11-finding-equation-line-point-slope-01
↑http://www.coolmath.com/algebra/08-lines/12-finding-equation-two-points-01
↑https://www.mathwarehouse.com/algebra/linear_equation/write-equation/equation-of-line-given-two-points.php
↑https://www.mathwarehouse.com/algebra/linear_equation/write-equation/equation-of-line-given-two-points.php
↑https://www.mathwarehouse.com/algebra/linear_equation/write-equation/equation-of-line-given-two-points.php
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