Linear equations come in three basic forms: point-slope, standard and slope-intercept. The general format of slope-intercept is y = Ax + B, where A and B are constants. Although the different forms are equivalent, providing the same results, the slope-intercept form quickly gives you valuable information about the line it produces.
TL;DR (Too Long; Didn't Read)
TL;DR (Too Long; Didn't Read)
The slope-intercept form of a line is y = Ax + B, where A and B are constants and x and y are variables.
Slope-Intercept Breakdown
The slope-intercept form, y = Ax + B has two constants, A and B, and two variables, y and x. Mathematicians call y the dependent variable because its value depends on what happens on the other side of the equation. The x is the independent variable because the rest of the equation depends on it. The constant A determines the slope of the line and B is the value of the y-intercept.
Slope and Intercept Defined
The slope of a line reflects the line’s “steepness,” and if it increases or decreases. To give some examples, a horizontal line has a slope of zero, a gently rising line has a slope with a small numeric value, and a steeply rising line has a slope with a large value. The fourth type of slope is undefined; it is vertical. The sign of the slope shows whether the line rises or falls in value going from left to right. A positive slope means the line rises, and a negative slope means it falls.
The intercept is the point at which the line crosses the y-axis. Going back to the form, y = Ax + B, you can find the point by taking the value of B and finding that number on the y axis, where x is zero. For example, if your line equation is y = 2x + 5, the point lies at (0, 5), right on the y axis.
Two Other Forms
In addition to the slope-intercept form, two other forms are in common use, standard and point-slope. The standard form of a line is Ax + By = C, where A, B and C are constants. For example, 10x + 2y = 1 describes a line in this form. The point-slope form is y − A = B(x − C). This equation provides an example of the point slope form:
Graphing with Slope-Intercept
You need two points to draw a line on a graph. The slope-intercept form gives you one of those points automatically — the intercept. Plot the first point using the value of B following the directions described above. Finding the second point takes a little algebra work. In your line equation, set the value of y to zero, then solve for x. For example, using
solve 0 = 2x + 5 for x:
Subtracting 5 from both sides gives you
Dividing both sides by 2 gives you
Mark the point at ( −5/2, 0). You already have a point at (0, 5). Using a ruler, draw a line connecting the two points.
Finding Parallel Lines
Creating a line parallel to one written as slope-intercept is simple. Parallel lines have the same slope but different y-intercepts. So simply keep the slope variable A from your original line equation and use a different variable for B. For example, to find a line parallel to
keep 3.5x and use a different number for B, such as 14, so the equation for the parallel line is
You may also need to find a line that passes through a particular point at (x, y). For this exercise, plug in the values of x and y and solve for the y-intercept, B. For example, you want to find the line that passes through the point (1, 1). Set x and y to the values of the point given and solve for B:
Substitute the point values for x and y:
Multiply the x value (1) by the slope (3.5):
Subtract 3.5 from both sides:
Plug the value of B into your new equation.
Finding Perpendicular Lines
Perpendicular lines cross one another at right angles. To do that, the slope of of the perpendicular line is −1 / A of the original line, or negative one divided by the original slope. To find a line perpendicular to
divide −1 by 3.5 and get the result, −2/7. Any line with the slope of −2/7 will be perpendicular to y = 3.5x + 20. To find a perpendicular line that passes through a given point (x, y), plug the values of x and y into your equation and solve for the y-intercept, B, as above.
References
About the Author
Chicago native John Papiewski has a physics degree and has been writing since 1991. He has contributed to "Foresight Update," a nanotechnology newsletter from the Foresight Institute. He also contributed to the book, "Nanotechnology: Molecular Speculations on Global Abundance." Please, no workplace calls/emails!