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Graphing Equations and Inequalities Survey A Complete each ordered pair. 1. ( 3, __ ) if 2x – 8y = -2 (3 , 1) 2. ( __ , -2 ) if y = x +7 (-24, -2) 3. ( -4 , __ ) if y = -3x – 5 (-4 , 7) 4. ( __ , -½ ) if 3x – 6y = -1 (-4/3 , -1/2) Survey A 2 Graphing Linear Equations Survey A Graph a linear equation by finding and plotting ordered pair solutions Survey A 4 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 are not both 0. This form is called standard form. The graph of a linear equation in two variables is a straight line. Survey A 5 Example 1 Graph the linear equation 2x – y = -4 To graph this equation, we find three ordered pair solutions by choosing a value for one of the variables, x or y, then solving for the other variable. (The third solution acts as a check for the other two.) We will plot the solution points, then draw the line containing the 3 points. Survey A 6 Graph the linear equation 2x – y = – 4. Let x = 1. 2x – y = –4 2(1) – y = –4 2 – y = –4 – y = –6 y=6 Replace x with 1. Simplify. Subtract 2 from both sides. Multiply both sides by –1. The ordered pair (1, 6) is a solution of 2x – y = – 4. Survey A 7 Graph the linear equation 2x – y = – 4. Next, let y = 4. 2x – y = –4 2x – 4 = –4 2x = – 4 + 4 2x = 0 x=0 Replace y with 4. Add 4 to both sides. Simplify. Divide both sides by 2. The ordered pair (0, 4) is a second solution. Survey A 8 Graph the linear equation 2x – y = – 4. Next, let x = -3. 2x – y = –4 2(-3) – y = –4 – 6 – y = –4 –y = 2 y = –2 Replace x with –3. Simplify. Add 6 to both sides. Multiply both sides by –1. The third solution is (–3, –2). Survey A 9 y (1, 6) Now we plot all three of the solutions (1, 6), (0, 4) and (–3, –2). Then, we draw the line that contains the three points. Survey A (0, 4) x (– 3, – 2) 10 Example 2 Graph the linear equation y = ¾x + 3. Since the equation is solved for y, we should choose values for x. To avoid fractions, we should select values of x that are multiples of 4 (the denominator of the fraction). Survey A 11 Graph the linear equation y = x + 3. Let x = 4. y = x + 3 y= (4) + 3 y=3+3=6 Replace x with 4. Simplify. So one solution is (4, 6). Survey A 12 Graph the linear equation y = x + 3. Next, let x = 0. y= y= x + 3 (0) + 3 y=0+3=3 Replace x with 0. Simplify. So a second solution is (0, 3). Survey A 13 Graph the linear equation y = x + 3. Next, let x = –4. y= x + 3 y= (–4) + 3 y = –3 + 3 = 0 Replace x with – 4. Simplify. So the third solution is (–4, 0). Survey A 14 y Now we plot all three of the ordered pair solutions; (4, 6), (0, 3) and (–4, 0). Then, we draw the line that contains the three points. Survey A (4, 6) (0, 3) x (–4, 0) 15 When graphing a linear equation in two variables, if it is: • solved for y, it may be easier to find ordered pair solutions by choosing x-values. If it is • solved for x, it may be easier to find ordered pair solutions by choosing y-values. Survey A 16 Example 3 Graph the linear equation y = 3 Can be written in standard form as 0x + y = 3. No matter what value we replace x with, y is always 3. x 0 1 5 Survey A y 3 3 3 17 1. Plot the points 2. Connect with line What kind of line is this? a horizontal line ** always y = # ** Survey A 18 Example 4 Graph the linear equation x = -2 Can be written in standard form as x + 0y = -2. No matter what value we replace y with, x is always -2. x -2 -2 -2 Survey A y -2 0 1 19 1. Plot the points 2. Connect with line What kind of line is this? a vertical line ** always x = # ** Survey A 20 What three things do I need to do in order to graph a linear equation? 1. Pick values to plug in to have ordered pairs 2. Plot the ordered pairs 3. Connect with a line Survey A 21