A linear equation is a fundamental concept in algebra. It's called linear because when we graph it on a coordinate plane, we get a straight line. The key characteristic is that all variables have an exponent of exactly one. For example, y equals 2x plus 1 is a linear equation. Notice how all the points that satisfy this equation lie perfectly on a straight line.
Linear equations have several standard forms. For one variable, we write ax plus b equals zero. For two variables, the general form is Ax plus By equals C. The most common form is slope-intercept form: y equals mx plus b, where m is the slope and b is the y-intercept. Each form has its advantages for different applications.
Linear equations have specific characteristics that distinguish them from other types of equations. All variables must have an exponent of exactly one. Variables cannot be multiplied together, appear in denominators, or under square roots. When graphed, linear equations always form straight lines. In contrast, non-linear equations like y equals x squared create curved graphs.
In the slope-intercept form y equals mx plus b, m represents the slope and b represents the y-intercept. The slope tells us how steep the line is and is calculated as rise over run. The y-intercept is where the line crosses the y-axis. For example, in y equals 1.5x plus 1, the slope is 1.5 and the y-intercept is 1.
Here are practical examples of linear equations. For one variable, we have equations like 2x plus 3 equals 7, which has the solution x equals 2. For two variables, examples include y equals 3x minus 5 and 4a minus 2b equals 10. We can even have three variables, like x plus y plus z equals 12. All these equations share the common property that their graphs form straight lines.