2. Linear Equation in Two Variables
A linear equation in two variables is an equation that can be written in the form
Ax + By = C
where A, B, and C are real numbers and A and B not both 0. The graph of a linear equation in two variables is a straight line.
The form Ax + By = C is called standard form.

4. Example Graph the linear equation 2x – y = – 4. Let x = 1.
2(1) – y = – Replace x with 1.
2 – y = – Simplify the left side.
– y = – 4 – 2 = – Subtract 2 from both sides.
y = Multiply both sides by – 1.
One solution to the equation is (1, 6).

5. Example continued: Graph the linear equation 2x – y = – 4.
For the second solution, let y = 4.
2x – 4 = – Replace y with 4.
2x = – Add 4 to both sides.
2x = Simplify the right side.
x = Divide both sides by 2.
The second solution is (0, 4).

6. Example continued: Graph the linear equation 2x – y = – 4.
For the third solution, let x = – 3.
2(– 3) – y = – Replace x with – 3.
– 6 – y = – Simplify the left side.
– y = – = Add 6 to both sides.
y = – Multiply both sides by – 1.
The third solution is (– 3, – 2).

7. x y
Example continued:
(1, 6)
(0, 4)
(– 3, – 2)
Now plot all three of the solutions (1, 6), (0, 4) and (– 3, – 2).
Draw the line with arrows that contains the three points.

10. Solution Example Graph y = 2
Writing in slope-intercept form: 0 • x + y = 2. No matter what number we choose for x, we find that y must equal 2.
y = 2
Choose any number for x x y
(x, y) 2
(0, 2) 4
(4, 2) 4
(4 , 2)
y must always be 2

11. Graph y = 2
When we plot the ordered pairs (0, 2), (4, 2) and (4, 2) and connect the points, we obtain a horizontal line.
Any ordered pair of the form (x, 2) is a solution, so the line is parallel to the x-axis.

12. Example Graph x = 2 Solution
We regard the equation x = 2 as x + 0 • y = 2. We make up a table with all 2 in the x-column.
x = 2
x must be 2 x y
(x, y) 2 4
(2, 4) 1
(2, 1) 4
(2, 4)
Any number can be used for y

13. Graph x = 2
When we plot the ordered pairs (2, 4), (2, 1), and (2, 4) and connect them, we obtain a vertical line.
Any ordered pair of the form (2, y) is a solution. The line is parallel to the y-axis.