2. Martin-Gay, Beginning Algebra, 5ed 22 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.

3. Martin-Gay, Beginning Algebra, 5ed 33 Graph the linear equation 2x – y = – 4. Let x = 1. 2(1) – y = – 4 Replace x with 1. 2 – y = – 4 Simplify the left side. – y = – 4 – 2 = – 6 Subtract 2 from both sides. y = 6 Multiply both sides by – 1. One solution to the equation is (1, 6). Example

4. Martin-Gay, Beginning Algebra, 5ed 44 Graph the linear equation 2x – y = – 4. For the second solution, let y = 4. 2x – 4 = – 4 Replace y with 4. 2x = – 4 + 4 Add 4 to both sides. 2x = 0 Simplify the right side. x = 0 Divide both sides by 2. The second solution is (0, 4). Example continued:

5. Martin-Gay, Beginning Algebra, 5ed 55 Graph the linear equation 2x – y = – 4. For the third solution, let x = – 3. 2(– 3) – y = – 4 Replace x with – 3. – 6 – y = – 4 Simplify the left side. – y = – 4 + 6 = 2 Add 6 to both sides. y = – 2 Multiply both sides by – 1. The third solution is (– 3, – 2). Example continued:

6. Martin-Gay, Beginning Algebra, 5ed 66 Now plot all three of the solutions (1, 6), (0, 4) and (– 3, – 2). x y (1, 6) (0, 4) (– 3, – 2) Draw the line with arrows that contains the three points. Example continued:

7. Martin-Gay, Beginning Algebra, 5ed 77

8. 88

9. 99 Since all points on the x-axis have a y-coordinate of 0, to find x-intercept, let y = 0 and solve for x Since all points on the y-axis have an x-coordinate of 0, to find y-intercept, let x = 0 and solve for y

10. Martin-Gay, Beginning Algebra, 5ed 10

11. Martin-Gay, Beginning Algebra, 5ed 11

12. Martin-Gay, Beginning Algebra, 5ed 12 Example Graph y = 2 Solution Writing in slope-intercept form: y = 0 x + 2. No matter what number we choose for x, we find that y must equal 2. Choose any number for x y must always be 2 xy(x, y) 02(0, 2) 42(4, 2) 44 2 (  4, 2) y = 2

13. Martin-Gay, Beginning Algebra, 5ed 13 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 with y-intercept (0, 2).

14. Martin-Gay, Beginning Algebra, 5ed 14 xy(x, y) 22 4 (  2, 4) 22 1 (  2, 1) 22 44(  2,  4) x must be  2 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. Any number can be used for y x =  2

15. Martin-Gay, Beginning Algebra, 5ed 15 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 with x-intercept (  2, 0).

16. Martin-Gay, Beginning Algebra, 5ed 16

17. Martin-Gay, Beginning Algebra, 5ed 17