Slope-Intercept Form of the Line Equation
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Slope-Intercept Form of the Line Equation

The slope-intercept form of a line’s equation, explicitly casts the variable y as a function of the variable x. The variable y appears all by itself on the left-hand side of the equation, while being absent from the right-hand side.

Slope-Intercept form equation: y equals mx plus b

The slope-intercept form of a line’s equation makes immediately apparent two of the most important features of a line in the XY-plane: its slope and its Y-intercept.

The slope is the m constant, the coefficient of x
The Y-intercept is the b constant, the equation’s independent term

In the applet below, you can change the numerical values of b and m, and watch how these changes impact the location of the green line:

The red slider on the bottom left controls the m value, the slope of the line; while the blue slider on the bottom right controls the b value, the line's y-intercept.

Moving the red slider makes the green line rotate around the point (0, b); while moving the blue slider makes the green line move up or down parallel to itself.

You can use this applet, and the other applet down below, to find the answers to many homework questions about straight lines given in the slope-intercept format.


The Y-intercept

The Y-intercept value b indicates the point (0, b), where the line crosses the Y-axis

  • When the Y-intercept is positive, the line crosses the Y-axis above the origin.
  • When the Y-intercept is negative, the line crosses the Y-axis below the origin.
  • When the Y-intercept is zero, the line passes through the origin.

For example, line { y = x + 3 } crosses the Y-axis at point (0, 3), that is 3 units above the origin;
while parallel line { y = x - 2 } crosses the Y-axis at point (0, -2), that is 2 units below the origin.

When two lines have different slopes but the same Y-intercept, they meet on the Y-axis, because them both cross the Y-axis at the exact same point (0, b).

A Formula for the Y-intercept Given the Coordinates of Two Points on the Line

If a line with slope-intercept equation { y = mx + b } passes through two points P1 = (x1, y1), and P2 = (x2, y2),
you can use the following formula to calculate b, the line's y-intercept:

Formula for the y-intercept of a line passing through two given points with coordinates (x1, y1) and (x2, y2)

For example, given the points (x1, y1) = (6,1) and (x2, y2) = (-2, -3)

Example of calculating the y-intercept b from coordinates (6, 1) and (-2, -3)


Slope-Intercept Interactive Applet with Three Movable Lines

Each line in the applet below has its slope and y-intercept values controled not by sliders but by the point (m, b)
Each control point affects the position of the line of the same color, by means of that point's coordinates

  • To change the slope of a line, move sideways the point of the same color
  • To change the y-intercept of a line, move up or down the point of the same color

A curious property of the (m, b) control point representation used in the above applet, is this:

  • When the three control points are lined up along any non-vertical straight line, all three corresponding lines meet at a common point
  • When the three control points are lined up along any vertical line, all three corresponding lines become parallel, since they all have the same slope m


The slope

The slope indicates how fast a general point on the line goes up as it moves to the right.

  • When the slope is a positive number, the line goes up to the right, and down to the left.
  • When the slope is negative, the line goes down to the right, and up to the left.
  • When the slope is zero, the line is horizontal.
  • When the slope has an absolute value greater than 1, the acute angle the line makes with the X-axis is greater than 45º
  • When the slope has an absolute value smaller than 1, the acute angle the line makes with the X-axis is smaller than 45º
  • A line with slope equal to 1 makes a 45º angle with the positive direction of the X-axis
  • A line with slope equal to -1 makes a 135º angle with the positive direction of the X-axis
The farther away from zero the slope value is, the faster y changes in response to a change in x, for a general point (x, y) on the line. This means the line becomes steeper.

For example, in the line { y = 2x + 3 }, when x changes from 1 to 2, y changes from 5 to 7.
This means 2 units of change in y for each unit of change in x, according to the slope value m = 2.

Now, if we consider the line { y = 4x + 2 }, when x changes from 1 to 2, y changes from 6 to 10.
This is a change of 4 units in y for each unit of change in x, according to the slope value m = 4.

On the other hand, the closer to zero the slope is, the slower y changes in response to a change in x, for a general point (x, y) on the line. This means the line is less steep.

For example, in the line { y = 0.5x + 6 }, when x changes from 2 to 4, y changes from 7 to 8.
This means the variable x needs to go up by 2 units in order for the variable y to increase a single unit.
This line with a slope of 0.5 grows more slowly than a line with a slope of 1.


Differences of the slope-intercept form with other equation forms

In contrast with both the general form, and the standard form of the line’s equation, the slope-intercept form of the equation provides a unique representation of a line, because when two lines have either a different slope, or a different Y-intercept, they really are two different lines.

In other words, if two equations in slope-intercept form have different coefficients, they do represent different lines. This stands in sharp contrast to the general form of the equation, where you can have many equations that look very different from each other, because they have different coefficients,




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Last review: May 20, 2018