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Slope-intercept form one of a number of different forms of a linear equation. It is one of the most commonly used forms, and has the following structure:
y = mx + b
In the equation above, m is the slope, and b is the y-intercept. Slope intercept form is useful because it allows us to quickly identify the slope and y-intercept of a line, which in turn allows us to graph the line fairly easily.
Given a point and the slope of a line, you can write a linear equation in slope-intercept form.
Example
Given m = 3 and the point (3, 5), find b, and write the equation of the line in slope-intercept form.
y = mx + b | (5) = (3)(3) + b | b = 5 - 9 = -4 | y = 3x - 4 |
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Because we are given slope and a point on a line, we just need to find the y-intercept in order to write the equation of the line in slope-intercept form. We found the y-intercept by plugging the given values into the equation y = mx + b, then solving for b. Once the slope and y-intercept are known, writing the equation of the line just involves plugging the slope and y-intercept into m and b respectively.
How to find the slope and the y-intercept
Given at least two points on a line, the slope of the line can be found using the slope formula:
For example, given the that (1, 5) and (-2, 7) are points on the same line, the slope of the line can be found as follows:
The y-intercept can be found in a number of ways. If a graph is given, the point at which the line intersects the y-axis is the y-intercept. This occurs when x = 0. It follows that, given an equation, setting x equal to 0 and solving for y will yield the y-intercept of the line. For example, given the line 3x + 2y = 6, the y-intercept can be found by plugging 0 in for x, then solving for y.
3(0) + 2y = 6
2y = 6
y = 3
Thus, the y-intercept occurs at the point (0, 3).
Other forms of linear equations
Slope-intercept form is just one of a number of different forms of linear equations, albeit it is the most commonly used form. Depending on the context that a linear equation is being used in, certain forms can be more beneficial to use. Other commonly used forms include point-slope form and standard form.
Slope-intercept form is useful because the slope and y-intercept of the line can easily be read off the equation, which also makes it relatively easy to graph the line, since we can just start from the y-intercept and find another point on the line by counting the rise over run (slope) in the coordinate plane.
Point-slope form
Point-slope form can be generalized as:
y - y_{1} = m(x - x_{1})
In the equation, y_{1} and x_{1} indicate a point on the line that is not the y-intercept, while m is the slope. Point-slope form is useful when a point on the line and the slope of the line are known, since it enables us to write an equation for the line.
Standard form
The standard form of the equation of a line is
Ax + By = C
where A, B, and C are integers. Standard form is useful when we are trying to solve systems of linear equations. It can also be used to write the equation of vertical lines, something that cannot be done using slope-intercept or point-slope form.
Converting between different forms
It is relatively common to need to convert the equation of a line in standard or point-slope form to slope-intercept form.
Standard to slope-intercept form
Converting from standard form to slope-intercept form involves manipulating an equation in standard form so as to isolate y on one side of the equation such that the coefficient on the y is 1.
Example
Convert 2x - 5y + 6 = 0 to slope-intercept form.
5y = 2x + 6
y = 2x/5 + 6/5
The slope is therefore 2/5, and the y-intercept is at (0, 6/5).
Point-slope to slope-intercept form
Converting from point-slope form to slope-intercept form is relatively simple since both already include a slope, so all we need to do is convert the equation such that we can read off the y-intercept, rather than some other point on the line.
Example
Convert y - 7 = 1/2(x - 4) to slope-intercept form.
y = 1/2x - 4/2 + 7
y = 1/2x + 5
FAQs
The equation of the line is written in the slope-intercept form, which is: y = mx + b, where m represents the slope and b represents the y-intercept. In our equation, y = 6x + 2, we see that the slope of the line is 6.
How to solve in slope-intercept form? ›
To find the y-intercept, find where the line hits the y-axis. To find the x-intercept (which wasn't mentioned in the text), find where the line hits the x-axis. To find the slope, find two points on the line then do y2-y1/x2-x1 the numbers are subscripts. Hope that helped.
Why is it necessary to express the answer in slope-intercept form? ›
The slope-intercept formula is used to find the slope, the y-intercept, the x-intercept, or the equation of a straight line given the requisite parameters. There are different formulas available to find the equation of a straight line.
What is the answer to slope y-intercept? ›
The slope-intercept form of a line is: y=mx+b where m is the slope and b is the y-intercept. The y-intercept is always where the line intersects the y-axis, and will always appear as (0,b) in coordinate form.
How do you write a slope answer? ›
1 Expert Answer
The equation for the slope, m, is: m=rise/run=(y_{2}-y_{1})/(x_{2}-x_{1}). m=(2-0)/(1-(-1))=2/2=1, so the slope between these two points is 1. y-0=1(x+1) or y=1(x+1), which is the equation of the line in point-slope form.
What is an example of slope-intercept form? ›
To graph a linear equation in slope-intercept form, we can use the information given by that form. For example, y=2x+3 tells us that the slope of the line is 2 and the y-intercept is at (0,3). This gives us one point the line goes through, and the direction we should continue from that point to draw the entire line.
How do you solve for slope form? ›
In mathematical terms, the slope is the rate of change of y with respect to x. When dealing with linear equations, we can easily identify the slope of the line represented by the equation by putting the equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
What is the easiest way to find the slope-intercept form of a line? ›
Given two points on a line, we can write an equation for that line by finding the slope between those points, then solving for the y-intercept in the slope-intercept equation y=mx+b.
How do I write an equation into slope-intercept form? ›
To change the equation into slope-intercept form, write it in the form y = m x + b . You want to isolate the y, so the first step is to multiply both sides by 9. Then cancel out the 9's on the right side.
How to solve slope-intercept form with two points? ›
How to Find the Equation of a Line from Two Points
- Find the slope using the slope formula. ...
- Use the slope and one of the points to solve for the y-intercept (b). ...
- Once you know the value for m and the value for b, you can plug these into the slope-intercept form of a line (y = mx + b) to get the equation for the line.
The slope, or steepness, of a line is found by dividing the vertical change (rise) by the horizontal change (run). The formula is slope =(y₂ - y₁)/(x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line.
What is the intercept formula? ›
In Maths, an intercept is a point on the y-axis, through which the slope of the line passes. It is the y-coordinate of a point where a straight line or a curve intersects the y-axis. This is represented when we write the equation for a line, y = mx+c, where m is slope and c is the y-intercept.
What is slope-intercept form simplified? ›
Slope intercept form reads y = mx + b , where m is the slope (steepness) of the line, and b is the y-intercept, i.e., the value at which the line intersects the vertical axis. For example, y = -2x + 3 . Standard form reads Ax + By + C = 0 , where A, B, C are integers. For example, 2x + y - 3 = 0 .
How to find slope-intercept form on a graph? ›
We can write the slope-intercept equation from a graph. The point where the graph crosses the y-axis is our b-value. The slope is our m-value. Plug these into y=mx+b.
What is the point slope formula? ›
Point-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). We can rewrite an equation in point-slope form to be in slope-intercept form y=mx+b, to highlight the same line's slope and y-intercept.
How do I write something in slope-intercept form? ›
Slope-intercept form (y=mx+b) of linear equations highlights the slope (m) and the y-intercept (b) of a line. Watch this video to learn more about it and see some examples.
How to rewrite an equation in slope-intercept form? ›
The equation of a line written in the form y = mx + b is said to be in slope-intercept form. To write an equation in slope-intercept form, you need to isolate y by using the properties of equality. Example: Rewrite the equation 4x - 2y = 12 in slope-intercept form. 4x-2y= 12 -4x -4x -2y = -4x + 12 -2 -2 -2 y=2x-6 1.
How to write an equation in slope-intercept form that passes through given points? ›
y = mx + b Calculate the slope, m, from the given point values, solve for b by using one of the point values, and check your solution using the other point values. A line can be thought of as the ratio of the change between horizontal (x) and vertical (y) positions.