A linear equation is an equation that describes a straight line on a graph. You can remember this by the "line" part of the name linear equation.
Standard Form
Linear equations have a standard form that looks like this:
Ax + By = C
Where A and B are coefficients (numbers) while x and y are variables. C is a constant.
You can think of the x and y variables as points on a graph.
Example linear equations:
You can plug numbers into A, B, and C of the above standard form to make linear equations:
2x + 3y = 7 x + 7y = 12 3x - y = 1
Linear Equations Represent Lines
At first it may seem strange that an equation represents a line on a graph. To make a line you need two points. Then you can draw a line through those two points.
The x and y variables in the linear equation represent the x and y coordinates on a graph. If you plug in a number for x, you can calculate the corresponding number for y. Those two numbers show a point on a graph. If you keep plugging in numbers for x and y in a linear equation, you will find that all the points together make a straight line.
Graphing a Linear Equation
In order to graph a linear equation you can put in numbers for x and y into the equation and plot the points on a graph. One way to do this is to use the "intercept" points. The intercept points are when x = 0 or y = 0. Here are some steps to follow:
Plug x = 0 into the equation and solve for y
Plot the point (0,y) on the y-axis
Plug y = 0 into the equation and solve for x
Plot the point (x,0) on the x-axis
Draw a straight line between the two points
You can check your answers by trying other numbers in the equation. Try x = 1. Solve for y. Then make sure that that point is on your line.
Example Problem:
Graph the linear equation: 2x + y = 2
Step 1: Plug in x = 0 and solve for y.
2 (0) + y = 2 y = 2
Step 2: Plug in y = 0 and solve for x.
2x + 0 = 2 2x = 2 x = 1
Step 3: Graph the x and y intercept points (0, 2) and (1,0)
Step 4: Draw a straight line through the two points
Step 5: Check the answer.
We will put in 2 for x and solve:
2(2) + y = 2 4 + y = 2 y = 2 - 4 y=-2
Is the point (2,-2) on the line?
You can try some other points to double check as well.
Example 2:
Graph the linear equation x - 2y = 2
Step 1: x = 0
0 - 2y = 2 y = -1
Step 2: y = 0
x - 2(0) = 2 x = 2
Step 3: Graph the x and y points (0, -1) and (2,0)
