Return to Free Math Homework Help   >>   Return to Algebra Main Page   >>   Return to Functions Main Page

Free Homework Math Help: Linear Functions

Linear Equations

Much of algebra has to do with manipulation and solving linear equations. Linear equations are equations of the line. In algebra, it is represented in one of two forms: standard form or slope-intercept form. In standard form, a line is represented as follows: Ax+Bx+C = 0 where A and B are coefficients and C is a constant while x and y are variables that correspond to the x and y axes, respectively. The second form is the slope intercept form and is represented as follows: y = mx+b where m is the slope and b is the y-intercept. The slope is the amount by which the line increasess or decreases over a given horizontal distance. A slope can be positive or negative. A positve slope is increasing and a negative slope is decreasing. The y-intercept is the place where the line intersects with the y-axis.

Point Slope Form of a Linear Equation

y-y0 = m(x-x0)

The y-intercept is the point (0,b).

Slope

Another way of saying slope is rise over run.

Slope Formula

The formula for the slope is: m   =   y2 - y1
x2 - x1

A horizontal line has zero slope, m = 0.

A vertical line has undefined slope.

A line with positive slope increases from left to rightA line with negative slope decreases from left to right

If a line has a certain slope m1, and another line is parallel to it with slope m2, then m1 = m2. If a line has slope m then a line perpendicular to it has a slope -1
m		.

Parallel lines have the same slope Perpendicular lines have negative inverse slope to each other

Properties, Rules and Formulas

The properties of linear equations are as follows:

x and y are function variables.

x0, x1, x2, y0, y1, y2 are coordinate values for points.

General Form Ax + By + C = 0
Point Slope Form y-y0 = m(x-x0)
Slope-Intercept Form        y = mx+b
Intercept Form
x
a		   +   y
b		   = 1
Two Point Form
y-y1   =   (y2 - y1)
(x2 - x1)		(x-x1)
Slope Formula
m   =   (y2 - y1)
(x2 - x1)
Vertical Line x = a
Horizontal Line y = b

An equation that can be written in the above forms is a linear equation.

The linear equation ax+b=0 has the solution x   =   -b
a		.

To solve a linear equation for an x or y answer, either of the variables has to be on one side of the equation or the other by itself. To isolate the variable being solved for, use the order of operator precedence, PEMDAS, starting from the lowest order to the highest. For example, to solve for y in Ax+By+C = 0, subtract Ax and C from both sides to get: By = -Ax-C.

Then divide both sides by B to get y   =   -A
B		x   -   C
B		.