The slope of a line also called the gradient of a line is a number that describes how "steep" it is. In the figure above press 'reset'.
Notice that for every increase of one unit to the right along the horizontal x-axis, the line moves down a half unit. It therefore has a slope of To get from point A to B along the line, we have to move to the right 30 units and down Again, this is a half unit down for every unit across.
Because the line slopes downwards to the right, it has a negative slope. As x increases, y decreases. If the line sloped upwards to the right, the slope would be a positive number.
Adjust the points above to create a positive slope. A way to remember this method is "rise over run". It is the "rise" - the up and down difference between the points, over the "run" - the horizontal run between them. Just remember that rise going downwards is negative. Positive slope Here, y increases as x increases, so the line slopes upwards to the right. The slope will be a positive number.
Negative slope Here, y decreases as x increases, so the line slopes downwards to the right. The slope will be a negative number. The line on the right has a slope of about Zero slope Here, y does not change as x increases, so the line in exactly horizontal.
The slope of any horizontal line is always zero. The line on the right goes neither up nor down as x increases, so its slope is zero. Undefined slope When the line is exactly vertical, it does not have a defined slope.How to Find Slope, Y-Intercept and Graph a Line
The two x coordinates are the same, so the difference is zero. The slope calculation is then something like When you divide anything by zero the result has no meaning. The line above is exactly vertical, so it has no defined slope.
We say "the slope of the line AB is undefined". For more on this see Slope of a vertical line. In the figure above click on "show angle".
By convention the angle is measured from any horizontal line parallel to x-axis. Lines with a positive slope up and to the right have a positive angle, and a negative angle for a negative slope. Change the slope by dragging A or B and see this for yourself. In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculatioons to be slightly off.
Home Contact About Subject Index. Definition: The slope of a line is a number that measures its "steepness", usually denoted by the letter m.Slope, sometimes referred to as gradient in mathematics, is a number that measures the steepness and direction of a line, or a section of a line connecting two points, and is usually denoted by m.
Generally, a line's steepness is measured by the absolute value of its slope, m. The larger the value is, the steeper the line. Given mit is possible to determine the direction of the line that m describes based on its sign and value:. Slope is essentially change in height over change in horizontal distance, and is often referred to as "rise over run.
In the case of a road the "rise" is the change in altitude, while the "run" is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth's curvature should be considered as a factor. The slope is represented mathematically as:. The above equation is the Pythagorean theorem at its root, where the hypotenuse d has already been solved for, and the other two sides of the triangle are determined by subtracting the two x and y values given by two points.
Given the points 3,4 and 6,8 find the slope of the line, the distance between the two points, and the angle of incline:.
While this is beyond the scope of this calculator, aside from its basic linear use, the concept of a slope is important in differential calculus. For non-linear functions, the rate of change of a curve varies, and the derivative of a function at a given point is the rate of change of the function, represented by the slope of the line tangent to the curve at that point.
Correct Answer :. Let's Try Again :. Try to further simplify. Ski Vacation? Parallel lines have the same slope, to find the parallel line at a given point you should simply calculate the Sign In Sign in with Office Sign in with Facebook.
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Please try again using a different payment method. Subscribe to get much more:.By Deborah J. A regression line is simply a single line that best fits the data in terms of having the smallest overall distance from the line to the points.
Statisticians call this technique for finding the best-fitting line a simple linear regression analysis using the least squares method. The slope of a line is the change in Y over the change in X. For example, a slope of. The y-intercept is the value on the y-axis where the line crosses. The coordinates of this point are 0, —6 ; when a line crosses the y- axis, the x- value is always 0. You may be thinking that you have to try lots and lots of different lines to see which one fits best.
The standard deviation of the x values denoted s x. The standard deviation of the y values denoted s y. You simply divide s y by s x and multiply the result by r.
Note that the slope of the best-fitting line can be a negative number because the correlation can be a negative number. A negative slope indicates that the line is going downhill. For example, if an increase in police officers is related to a decrease in the number of crimes in a linear fashion; then the correlation and hence the slope of the best-fitting line is negative in this case.
The correlation and the slope of the best-fitting line are not the same. The formula for slope takes the correlation a unitless measurement and attaches units to it. Think of s y divided by s x as the variation resembling change in Y over the variation in X, in units of X and Y. For example, variation in temperature degrees Fahrenheit over the variation in number of cricket chirps in 15 seconds. So to calculate the y -intercept, bof the best-fitting line, you start by finding the slope, m, of the best-fitting line using the above steps.
Then to find the y- intercept, you multiply m by. Always calculate the slope before the y- intercept. The formula for the y- intercept contains the slope! Deborah J. How to Calculate a Regression Line. Scatterplot of cricket chirps in relation to outdoor temperature. About the Book Author Deborah J.Would you like to know how to calculate the equation of a line?
What data do you need to calculate it? How to interpret the equation of a line? One of the biggest problems you usually have in finding the line equation is that most of the time it is not clear what form this equation takes. Therefore, it is not known what each of the terms of the equation means, much less how to calculate it. There are other ways of calculating the equation of a line, but what I am going to teach you next will serve as a starting point and will get you out of more than one predicament.
For me it is the easiest way to calculate the equation of a line and you will also be able to solve all the exercises that you need to calculate lines, except those that tell you to calculate a certain form of equation. This is the explicit equation of a line.
In the first member we have the clear y and in the second member we have two terms, one with x multiplied by the coefficient m and another term formed by the coefficient n. I just want you to have a global vision. The slope has to do with the inclination of the line with respect to the x-axis. The steeper the slope, the more inclined the line and the less inclined the slope. It can be positive or negative. There are many ways to calculate the slope of a line.
As mentioned before, the slope indicates the inclination of the line with respect to the x-axis. This slope is calculated by dividing the vertical distance by the horizontal distance between two points on a line. Those two points of a line, in general will have coordinates x1,y1 and x2,y2 for points 1 and 2 respectively:. The vertical distance is calculated by subtracting the y-axis coordinates from each point, and the horizontal distance by subtracting the x-axis coordinates from each point.
Therefore m can be calculated with this formula:. Another way to calculate the slope, if this angle is known is with the tangent:. Another way to obtain the slope of a line is to indicate that it is parallel or perpendicular to another given line. Parallel lines have the same slope.
Perpendicular lines have this relationship between their slopes:. Therefore, the first thing to do in these cases is to calculate the slope of the given line.
To do this, we must put it in its explicit form:. Y the quotient m, will be the slope of the line, that is, the number that is multiplied by x. I remind you that if it has nothing, it is equivalent to having a 1. Any line parallel to the previous one will have the same slope, i. Once we have defined the slope of the given line, the slope of the perpendicular line, keeps this relation as we have seen before:.
Well, we already know how to calculate slope, but slope alone is not enough to calculate the equation of a line. Rectas with the same slope are infinite. Therefore, to define the line we want exactly, we also need to know a point where the line we want to calculate passes. Therefore, to calculate the equation of a line we need to know the slope and a point through which it passes.In mathematicsa linear equation is an equation that may be put in the form.
The coefficients may be considered as parameters of the equation, and may be arbitrary expressionsprovided they do not contain any of the variables. Alternatively a linear equation can be obtained by equating to zero a linear polynomial over some fieldfrom which the coefficients are taken.
The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true. Often, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown. In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables.
This is the origin of the term linear for describing this type of equations. Linear equations occur frequently in all mathematics and their applications in physics and engineeringpartly because non-linear systems are often well approximated by linear equations. This article considers the case of a single equation with coefficients from the field of real numbersfor which one studies the real solutions. All of its content applies to complex solutions and, more generally, for linear equations with coefficients and solutions in any field.
For the case of several simultaneous linear equations, see system of linear equations. In this case, the name unknown is sensibly given to the variable x. Either b equals also 0, and every number is a solution. In this latter case, the equation is said to be inconsistent. These equivalent variants are sometimes given generic names, such as general form or standard form. There are other forms for a linear equation see belowwhich can all be transformed in the standard form with simple algebraic manipulations, such as adding the same quantity to both members of the equation, or multiplying both members by the same nonzero constant.
It has therefore a unique solution for ywhich is given by. This defines a function. However, in linear algebraa linear function is a function that maps a sum to the sum of the images of the summands. For avoiding confusion, the functions whose graph is an arbitrary line are often called affine functions. With this interpretation, all solutions of the equation form a lineprovided that a and b are not both zero. Conversely, every line is the set of all solutions of a linear equation.
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. There are various ways of defining a line.
The Slope of a Straight Line
In the following subsections, a linear equation of the line is given in each case. A non-vertical line can be defined by its slope mathematics mand its y -intercept y 0 the y coordinate of its intersection with the y -axis. In this case its linear equation can be written. If, moreover, the line is not horizontal, it can be defined by its slope and its x -intercept x 0.
In this case, its equation can be written. These forms rely on the habit of considering a non vertical line as the graph of a function. In this case, a linear equation of the line is.Lines Par. One of the most important properties of a straight line is in how it angles away from the horizontal. This concept is reflected in something called the "slope" of the line. Its graph looks like this:. Slope of a Line. I'll pick two x -values, plug them into the line equation, and solve for each corresponding y -value.
By the way, I picked those two x -values precisely because they were multiples of three; by doing so, I knew I'd be able to clear the denominator of the fraction so I'd end up with nice, neat integers for my resulting y -values. It's not a rule that you have to do that, but it's a helpful technique.
To find the slope, designated by " m ", we can use the following formula:. Why " m " for "slope", rather than, say, " s "? The official answer is: Nobody knows. In case you haven't encountered those lower-than-the variables numbers before, they're called "subscripts".
Subscripts are commonly used to differentiate between similar things, or to count off, for instance, in sequences. In the case of the slope formula, the subscripts merely indicate that we have a "first" point whose coordinates are subscripted with a " 1 " and a "second" point whose coordinates are subscripted with a " 2 ". In other words, the subscripts indicate nothing more than the fact that we have two points that we're working with. It is entirely up to you which point you label as "first" and which you label as "second".
As logic dictates, the angle of the line isn't going to change just because you looked at the two points in a different order.
For computing slopes with the slope formula, the important thing is that we are careful to subtract the x 's and y 's in the same order. For our two points, if we choose 3, —2 to be our "first" point, then we get the following:. The first y -value above, the —2was taken from the point 3, —2 ; the second y -value, the 2came from the point 9, 2 ; the x -values 3 and 9 were taken from the two points in the same order.
If, on the other hand, we had taken the coordinates from the points in the opposite order, the result would have been exactly the same value:. As you can see, the order in which you list the points really doesn't matter, as long as you subtract the x -values in the same order as you subtracted the y -values. Because of this, the slope formula can be written as it was above, or alternatively it can also be written as:.