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Linear Equations in Algebra

by Ron Kurtus (revised 31 August 2015)

A linear equation is an equation where each variable is of the first-order and there is no multiplication of variables. (x is a first-order variable and x2 is a second-order variable.)

Linear equations graph as straight lines. Common linear equations in Algebra are those with one, two or three variables.

You can graph the equations by first making a table where you substitute values and then plotting out the graph. A linear equation with one variable has one solution, but linear equations of two and more variables have an infinite number of possible solutions.

Questions you may have include:

This lesson will answer those questions.



Equation with one variable

The standard or ideal form of a linear equation with one variable is ax + b = 0, where a and b are constants, x is the variable, and a is not equal to 0.

You can solve the equation for x to get x = − b/a.

(See Solving a Linear Equation with One Variable for more information.)

Since x = − b/a is a linear equation, it will graph as a straight line. For example, consider the equation 2x − 6 = 0. Its x-y graph is seen below:

Graph of linear equation with one variable

Graph of linear equation with one variable

With two variables

The standard form of a linear equation with two variables is y = ax + b, provided a is not equal to 0. Since this is a linear equation, it also graphs as a straight line.

Consider the equation y = 2x + 1. You can graph this equation by first making a table and inserting values for x and y. This also shows that there are numerous solutions to the equation (actually and infinite number of solutions).

For example, when x = 1, y = 2*1 + 1 = 3. The table of y = 2x + 1 values is as follows:

   x    y = 2x + 1
0
1
1
3
2
5
3
7

Table of y = 2x + 1 values

The graph of the equation is then:

Graph of linear equation with two variables

Graph of linear equation with two variables

With three or more variables

A linear equation with three variables would be in the form z = ax + by + c, provided a and b are not equal to 0. This equation would also be a straight line on a graph and have an infinite number of solutions. The graph would have three axes and thus be a 3-dimensional graph, which is difficult to draw.

w = ax + by + cz +d would be the form of a linear equation with four variables, provided a, b and c are not equal to 0. Although in theory, the graph would be a straight line, it would need to be drawn in the 4th dimension, and thus be not possible for humans to do.

The mathematical logic can be followed for linear equations with an even greater numbers of variables.

Summary

A linear equation is an equation that graphs as a straight line. Common linear equations are those with one, two or three variables. Each variable is of the first order and there is no multiplication of variables. You can graph the equations. A linear equation with one variable has one solution and those with two and more variables have an infinite number of possible solutions.


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