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.
How to Find the Equation of a Straight LineQuestion: 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 EquationsThe 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
NoteNEGATIVE 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 EQUATIONSThe 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):
CONSTANTSSome 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: