Created by Álvaro Díez and Bogna Szyk Show
Reviewed by Dominik Czernia, PhD candidate and Jack Bowater Last updated: Oct 06, 2022 The slope intercept form calculator will teach you how to find the equation of a line from any two points that this line passes through. It will help you to find the coefficients of slope and y-intercept, as well as the x-intercept, using the slope intercept formulas. Read on to learn what is the slope intercept form of a linear equation, how to find the equation of a line and the importance of the slope intercept form equation in real life. What is the slope intercept form?Any line on a flat plane can be described mathematically as a relationship between the
vertical (y-axis) and horizontal (x-axis) positions of each of the points that contribute to the line. This relation can be written as In this slope intercept calculator, we will focus only on the straight line. You can check our average rate of change calculator to find the relation between the variables of non-linear functions. Linear equations, or straight-line equations, can be quickly recognized as they have no terms with exponents in them. (For example, you will find an As we have seen before, you can write the equation of any line in the form of The term slope is the incline, or gradient, of a line. It tells us how much The y-intercept is the value of Slope intercept formula derivationStill need to know how to find the slope intercept form of a linear equation? We will assume you know two points that the straight line goes through. The first one will have coordinates (x₁, y₁) and the second one (x₂, y₂). Your unknowns are the
slope Firstly, substitute the coordinates of the two points into the slope intercept equation: (1) y₁ = mx₁ + b (2) y₂ = mx₂ + b Then, subtract the first equation from the second: y₂ - y₁ = m(x₂ - x₁) Finally, divide both sides of the equation by (x₂ - x₁) to find the slope: m = (y₂ - y₁)/(x₂ - x₁) Once you have found the slope, you can substitute it into the first or second equation to find the y-intercept: y₁ = x₁(y₂ - y₁)/(x₂ - x₁) + b b = y₁ - x₁(y₂ - y₁)/(x₂ - x₁) How to find the equation of a line?This slope intercept form calculator allows you to find the equation of a line in the slope intercept form. All you have to do is give two points that the line goes through. You need to follow the procedure outlined below.
Find the x-intercept and y-interceptIt is also always possible to find the x-intercept of a line. It is the value of x at which the straight line crosses the x-axis (it means the value of
Our slope intercept form calculator will display both the values of the x-intercept and y-intercept for you. Still, if you would like to learn more about them, we recommend you visit our x- and y-intercept calculator. Real world uses of y-intercept and x-interceptWe have already seen what is the slope intercept form, but to understand why the slope intercept form equation is so useful, you should know some applications it has in the real world. Let's see a couple of examples. We will start with simple ones from physics so that you can get an intuitive idea of what the y-intercept and x-intercept mean. Imagine a car moving at a fixed speed toward you. Its movement can be plotted as time versus the distance the car is from you (as shown above). This means that the x-axis will represent the time passed, and the y-axis will represent the distance to the car. You can even imagine the car
has started to move before you started the timer (that is: before Now, if you look at the y-intercept ( Looking now at the
x-intercept ( Other equations with y-interceptThe car example above is a very simple one that should help you understand why the slope intercept form is
important and, more specifically, the meaning of the In fact, the example above does not fit a linear equation and still has both intercepts. The same is true for any other parabola or another shape. One equation that is guaranteed to have a y-intercept but not necessarily an x-intercept is a parabola. This is equation is shown in the image above. It has a maximum or a minimum (depending on the orientation). If this maximum is below the x-axis or the minimum is above the x-axis, there will never be an x-intercept. However, unlike humans, not all equations are equal. Some of the formulas describe curves that might never intercept the x-axis, the y-axis, or both. Let's see in a bit more detail how this can be. Equations with no intercept (asymptote)We can distinguish 3 groups of equations depending on whether they have a y-intercept only, an x-intercept only, or neither. The first group (y-intercept only) can have almost any type of equation, including linear equations. A good easy example is The second and third groups of equations are a bit more tricky to imagine and to understand them well, we need to introduce the concept of an asymptote. An asymptote is a line (that can be expressed as a linear equation) to which the function or curve we are talking about gets closer and closer but never actually crosses or touches that line. The definition might not seem totally clear, but if we look at an example equation, we will have fewer problems with understanding it. Let's take the equation If we take values closer and closer to In this case, the linear equation In fact, the example we have shown you ( Before we move to our next topic, it is important to note that we have made extreme over-simplifications when talking about infinity, but we feel it is a good and fast approach for those that are not used to the concept of working with infinity in math. We recommend that you learn more about the proper ways of infinity, starting with the undefined expressions in math. Intercepts and linear equations in machine learning and scienceOne could easily think that the usefulness of linear equations is very limited due to their simplicity. However, the reality is a bit different. Linear equations are at the core of some of the most powerful methods to solve minimization and optimization problems. Minimization problems are a type of problem in which one would like to find how to make one of the variables as small as possible. This variable could be, for example, the difference between a prediction made by a model and reality. These types of problems are one of the most common problems and are at the core of machine learning and scientific experiments. One of the most common and powerful methods to find the minimum value of an equation or formula is the so-called Newton method, named after the genius that invented it. The way it works is by using derivatives, linear equations, and x-intercepts: Example of using the Newton method (Source: Wikimedia)This method consists of choosing a value of Once the x-intercept is calculated, that value of One very common example is when using the chi-square method to fit some data to a formula or trend. In this case, the value that we want to minimize is the sum of the squared distance from the trend line to the data points, where the distance is calculated along a perpendicular line from the point to the trend line. Álvaro Díez and Bogna Szyk Slope intercept form: y = mx + b Average rate of changeBilinear interpolationCatenary curve… 35 more How do you convert slope intercept to standard form?The standard form of a linear equation is Ax+By=C. To change an equation written in slope-intercept form (y=mx+b) to standard form, you must get the x and y on the same side of the equal sign and the constant on the other side. Use inverse operations to move terms.
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