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.
SlopeIntercept 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 slopeintercept form :
ax + by + c = 0
ax + by = − c
by = − ax − c
y = (− a / b ) x − c / b
y = (− a / b ) x + (− c / b )
where a, b , and c are realnumber 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 yvalue) of the point at which the line crosses the y axis; this is called the yintercept.
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 slopeintercept 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:
 Convert the equation to slopeintercept form
 Plot the point y = k
 Move to the right by n units on the graph
 If m is positive, move upward mn units
 If m is negative, move downward mn units, where m is the absolute value of m
 If m = 0, don’t move up or down at all
 Plot the resulting point y = mn + k
 Connect the two points with a straight line
Figures 65A and 65B illustrate the following linear equations as graphed in slopeintercept form:
y = 5 x − 3
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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