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3 .2 Graphing Linear Equations Ordered Pairs as Solutions to Equations An ordered pair is a solution to an equation in two variables, x and y , if the given x – value and y – value, when substituted into the equation, yield a true statement. Example (a) Is (2, 3) a solution to y = 2 x – 1? Given an x – value or a y – value, we can complete the ordered pair that satisfies an equation in x and y by substituting the given value for the given variable and solving for the other variable. Example (b) For y = – 3 x + 4, complete the ordered pair, (0, ). The completed ordered pair is ( 0 , 4 ) Demonstration Problems Practice Problems 1. (a) Determine if (1, 5) is a solution to x – y = – 4 2. (a) Complete the ordered pair (4, ) y = – 3 x + 5 1. (b) Determine if (2, 1) is a solution to x + y = – 3 2. (b) Complete the ordered pair (1, ) y = 2 x – 4 Answers: 1. (b) No; 2. (b) (1, – 2)
y = 2 x – 1 3 = 2 • 2 – 1? 3 = 4 – 1? 3 = 3 P ( 2 , 3 ) is a solution to y = 2 x – 1 because 3 = 2 • 2 – 1 is a true statement as shown at right. ( 0 , ) y = – 3 x + 4 y = – 3 • 0 + 4 y = 0 + 4 y = 4 To complete the ordered pair substitute 0 for x and solve for y as shown as right.
Completing a Table of Values Completing a table of values is the same as completing a set of ordered pairs. Use the method described on the previous page to complete the tables that follow. Demonstration Problems Practice Problems 3. (a) € y =
x − 1 x y
- 2 0 2 4 4. (a) 2 x – 3 y = 6 x y
- 3 0 3 6 3. (b) y = 2 x – 4 x y
- 2
- 1 0 1 4. (b) x + 4 y = 4 x y
- 4 0 4 8 Answers: 3. (b) – 8, – 6, – 4, – 2; 4. (b) 2, 1, 0, – 1;
Using a Table of Values to Graph Equations that Contain Fractions In making a table of values for a linear equation, we are free to choose any suitable x values. If the coefficient of x is a fraction, it is best to choose x values that are multiples of the denominator of the fraction. Example (d) Graph 2 x – 3 y = 6 Step 1 : Solve for y. 2 x – 3 y = 6 Step 2 : Complete the table of values. x y
- 6
- 3 0 3 6 Step 3 : Largest x : Smallest x : Largest y : Smallest y : Draw axes that accommodate these values. Step 4 : Plot the ordered pairs from the table of values and draw a line through the points. Choose x - values that are multiples of 3.
Using X -‐Intercepts and Y -‐Intercepts to Graph Lines The x-‐intercept is the point at which the line crosses the x -‐axis. The y-‐intercept is the point at which the line crosses the y -‐axis. Example (e) Graph 6 x – 3 y = 12 Instead of completing a table of values of many ordered pairs, we can find the x - intercept and y - intercept of the line, plot the two points and draw a line through them. Step 1 : Complete the table of values (to find the x - intercept and y - intercept) 6 x – 3 y = 12 x y x - intercept (^0) y - intercept (^0) Step 3 : Plot the two points from the table of values and draw a line through them. Step 2 : Largest x : Smallest x : Largest y : Smallest y : Draw axes that accommodate these values adjusting the scale where necessary. The x -‐coordinate of the y -‐intercept is 0. The y -‐coordinate of the x -‐intercept is 0.
Demonstration Problems Practice Problems 5. (a) Find the x -‐intercept and y -‐intercept, then use them to graph 2 x – 5 y = 10 6. (a) Graph x = 6 5. (b) Find the x -‐intercept and y -‐intercept, then use them to graph x + 4 y = 4 6. (b) Graph y = – 1 Answers: 5. (b) 4, 1;. 6. (b). x y x - intercept (^0) y - intercept (^0) x y x - intercept (^0) y - intercept (^0)
Interpreting a Graph as a Set of Ordered Pairs Given the graph of a linear equation, we can determine ordered pairs that are solutions to that equation by inspection. Example (h) Complete the ordered pairs using the graph at right. (i) (–4, ) Find – 4 on the x -‐axis and find the corresponding y -‐value to the line. The completed ordered pair is (– 4 , 1 ) (ii) ( , – 3) Find – 3 on the y -‐axis and find the corresponding x -‐value to the line. The completed order pair is ( 4 , – 3 ) Demonstration Problems Practice Problems Complete the ordered pairs using the graph above. 7. (a) (0, ) 8. (a) ( , 2) Complete the ordered pairs using the graph above. 7. (b) (2, ) 8. (b) ( , 0) Answers: 7. (b) – 2; 8. (b) – 2
