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.

Example 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 continued: 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: 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).

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.

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

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.

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

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.