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Point Slope and Slope Intercept Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Introduction to Forms of Linear Equations - Standard and Slope Form

Straight lines on the Cartesian plane are represented by a certain type of equation called a linear equation . There are several forms in which a linear equation can be written. All linear equations can be reduced to a form where neither x nor y is raised to any power other than 0 or 1.

Standard Form Of Linear Equation

The standard form of a linear equation in variables x and y consists of constant multiples of the two variables, plus another constant, all summed up to equal zero: 

ax + by + c = 0

In this equation, the constants are a, b , and c . If a constant happens to be equal to 0, then it is not written down, nor is its multiple (by either x or y ) written down. Examples of linear equations in the standard form are:

2 x + 5 y − 3 = 0

5 y − 3 = 0

2 x − 3 = 0

2 x = 0

5 y = 0

The last two of these equations can be simplified to x = 0 and y = 0, by dividing each side by 2 and 5, respectively.

Slope-Intercept Form Of Linear Equation

A linear equation in variables x and y can be manipulated so it is in a form that is easy to plot on the Cartesian plane. Here is how a linear equation in standard form can be converted to slope-intercept form :

ax + by + c = 0

ax + by = − c

by = − axc

y = (− a / b ) xc / b

y = (− a / b ) x + (− c / b )

where a, b , and c are real-number constants, and b ≠ 0. The quantity − a/b is called the slope of the line, an indicator of how steeply and in what sense the line slants. The quantity − c/b represents the ordinate (or y-value) of the point at which the line crosses the y axis; this is called the y-intercept.

What Is Slope?

Let dx represent a small change in the value of x on such a graph; let dy represent the change in the value of y that results from this change in x . The ratio dy/dx is the slope of the line, and is symbolized m . Let k represent the y -intercept. Then m and k can be derived from the coefficients a, b , and c as follows, provided b ≠ 0: 

m = − a/b

k = − c/b

The linear equation can be rewritten in slope-intercept form as:

y = (− a/b ) x + (− c/b )

and therefore:

y = mx + k

To plot a graph of a linear equation in Cartesian coordinates, proceed as follows:

  1. Convert the equation to slope-intercept form
  2. Plot the point y = k
  3. Move to the right by n units on the graph
  4. If m is positive, move upward mn units
  5. If m is negative, move downward |m|n units, where |m| is the absolute value of m
  6. If m = 0, don’t move up or down at all
  7. Plot the resulting point y = mn + k
  8. Connect the two points with a straight line

Figures 6-5A and 6-5B illustrate the following linear equations as graphed in slope-intercept form:

y = 5 x − 3

y = − x + 2

The Cartesian Plane Straight Lines Point-slope Form Of Linear Equation

The Cartesian Plane Straight Lines Point-slope Form Of Linear Equation

Fig. 6-5 . (A) Graph of the linear equation y = 5 x − 3. (B) Graph of the linear equation y = − x + 2.

A positive slope indicates that the line ramps upward as you move from left to right, and a negative slope indicates that the line ramps downward as you move from left to right. A slope of 0 indicates a horizontal line. The slope of a vertical line is undefined because, in the form shown here, it requires that m be defined as a quotient in which the denominator is equal to 0.

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