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What Is a Linear Equation? (Mathematics Lesson)

What Is a Linear Equation?

A linear equation is an equation that represents a straight line.

A linear equation is often written in the form:



On a graph, a linear equation looks like a straight line:



  • y is how far up the line is.

  • x is how far along the line is.

  • m is the slope, or gradient, of the line.

  • c is the y-intercept of the line, where it crosses the y-axis.
Note: This form of a linear equation is the slope-intercept form.

How to Find the Equation of a Straight Line

Question: What is the equation of the line drawn below?



  • What is the y-intercept, c?

    The y-intercept, c, is where the line crosses the y-axis.




  • What is the slope, m?

    The slope, or gradient, m, is how steep a line is.

    It is the change in height, divided by the change in horizontal distance.

    To find the slope of the graph, draw a right-angled triangle under the line. Divide the change in y by the change in x:



    A slope of 1 means the line goes up by 1 everytime it goes across by 1.

  • What is the linear equation?

    Insert the values of m and c into the standard form for a linear equation, y = mx + c:



    Note: The coefficient of the x is a 1. The 1 does not have to be written.

Other Forms of Linear Equations

The slope-intercept form is the most common form of a linear equation, but other forms exist.

The general form of a linear equation is:

Read more about the general form of a linear equation

The slope-point is another form of a linear equation:



m is the slope of the line, and the point (x1, y1) is a point on the line.

Read more about the slope-point form of a linear equation
Curriculum
Algebra Lessons
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Note
NEGATIVE SLOPES AND Y-INTERCEPTS

A slope can be negative. In this case, m is negative in the linear equation (e.g. y = -3x + 1) and the line slopes downwards:



A y-intercept can be negative. In this case, the c is negative in the linear equation (e.g. y = 2x - 3) and the line crosses the y-axis below the x-axis:



DIFFERENT FORMS OF LINEAR EQUATIONS

The many different forms of linear equations may be quite confusing. But they all have some things in common:

  • There are 2 variables, y and x.

  • There are constants, like 2 or c.

  • The y and x have a power of 1.

    You will not see other powers, such as 2:



    These are quadratic equations, not linear equations.

    Neither will you see powers of 3 (cubic equations) or higher.

    You will not see any square roots (where the power is ½):




CONSTANTS

Some lines don't have any slope - they go straight across or up:



  • The lines that go straight across always have the same value of y. The equation of the line is:



    where c is the y-intercept.

    Below are some examples:



  • The lines that go straight up always have the same value of x. The equation of the line is:



    where b is the x-intercept.

    Below are some examples: