5 4 Write Linear Equations in Standard Form

5. 4 Write Linear Equations in Standard Form Warm Up Lesson Presentation Lesson Quiz

5. 4 Warm-Up Write an equation in point-slope form of the line that passes through the given points. 1. (1, 4), (6, – 1) ANSWER y – 4 = –(x – 1) or y + 1 = –(x – 6) 2. ( – 1, – 2), (2, 7) ANSWER y + 2 = 3(x + 1) or y – 7 = 3(x – 2)

5. 4 Warm-Up 3. A store rents 3 DVDs for $5, plus $3 for each additional DVD. Find the cost of renting 20 DVDs. ANSWER $56

5. 4 Example 1 Write two equations in standard form that are equivalent to 2 x – 6 y = 4. SOLUTION To write one equivalent equation, multiply each side by 2. 4 x – 12 y = 8 To write another equivalent equation, multiply each side by 0. 5. x – 3 y = 2

5. 4 Guided Practice 1. Write two equations in standard form that are equivalent to x – y = 3. ANSWER Sample answer: 2 x – 2 y = 6, 3 x – 3 y = 9

5. 4 Example 2 Write an equation in standard form of the line shown. SOLUTION STEP 1 Calculate the slope. 1 – (– 2) 3 m= = – 1 = – 3 1– 2 STEP 2 Write an equation in point-slope form. Use (1, 1). y – y 1 = m(x – x 1) Write point-slope form. y – 1 = – 3(x – 1) Substitute 1 for y 1, 3 for m and 1 for x 1.

5. 4 Example 2 STEP 3 Rewrite the equation in standard form. 3 x + y = 4 Simplify. Collect variable terms on one side, constants on the other.

5. 4 Guided Practice 2. Write an equation in standard form of the line through (3, – 1) and (2, – 3). ANSWER – 2 x + y = – 7

5. 4 Example 3 Write an equation of the specified line. a. Blue line b. Red line SOLUTION a. b. The y-coordinate of the given point on the blue line is – 4. This means that all points on the line have a y-coordinate of – 4. An equation of the line is y = – 4. The x-coordinate of the given point on the red line is 4. This means that all points on the line have an x-coordinate of 4. An equation of the line is x = 4.

5. 4 Guided Practice Write equations of the horizontal and vertical lines that pass through the given point. 3. (– 8, – 9) ANSWER y = – 9, x = – 8 4. (13, – 5) ANSWER y = – 5, x = 13

5. 4 Example 4 Find the missing coefficient in the equation of the line shown. Write the completed equation. SOLUTION STEP 1 Find the value of A. Substitute the coordinates of the given point for x and y in the equation. Solve for A. Ax + 3 y = 2 A(– 1) + 3(0) = 2 –A = 2 A = – 2 Write equation. Substitute – 1 for x and 0 for y. Simplify. Divide by – 1.

5. 4 Example 4 STEP 2 Complete the equation. – 2 x + 3 y = 2 Substitute – 2 for A.

5. 4 Guided Practice Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation. 5. – 4 x + By = 7, (– 1, 1) ANSWER 6. 3; – 4 x + 3 y = 7 Ax + y = – 3, (2, 11) ANSWER – 7; – 7 x +y = – 3

5. 4 Example 5 LIBRARY Your class is taking a trip to the public library. You can travel in small and large vans. A small van holds 8 people and a large van holds 12 people. Your class could fill 15 small vans and 2 large vans. a. Write an equation in standard form that models the possible combinations of small vans and large vans that your class could fill. b. Graph the equation from part (a). c. List several possible combinations.

5. 4 Example 5 SOLUTION a. Write a verbal model. Then write an equation. 8 s + 12 p l = Because your class could fill 15 small vans and 2 large vans, use (15, 2) as the s- and l-values to substitute in the equation 8 s + 12 l = p to find the value of p. 8(15) + 12(2) = p Substitute 15 for s and 2 for l. Simplify. 144 = p Substitute 144 for p in the equation 8 s + 12 l = p.

5. 4 Example 5 ANSWER The equation 8 s + 12 l = 144 models the possible combinations. b. Find the intercepts of the graph. Substitute 0 for s. 8(0) + 12 l = 144 l = 12 Substitute 0 for l. 8 s + 12(0) = 144 s = 18

5. 4 Example 5 Plot the points (0, 12) and (18, 0). Connect them with a line segment. For this problem only nonnegative whole-number values of s and l make sense. c. The graph passes through (0, 12), (6, 8), (12, 4), and (18, 0). So, four possible combinations are 0 small and 12 large, 6 small and 8 large, 12 small and 4 large, 18 small and 0 large.

5. 4 Guided Practice 7. WHAT IF? In Example 5, suppose that 8 students decide not to go on the class trip. Write an equation that models the possible combinations of small and large vans that your class could fill. List several possible combinations. ANSWER 8 s + 12 l = 136; 17 small, 0 large; 14 small, 2 large; 11 small, 4 large; 8 small, 6 large; 5 small, 8 large; 2 small, 10 large

5. 4 Lesson Quiz Write an equation in standard form of the line that passes through the given point and has the given slope m or that passes through the two given points. 1. (1, – 6), m = – 2 ANSWER 2 x + y = – 4 2. (– 4, – 3), (2, 9) ANSWER – 2 x + y = 5

5. 4 3. Lesson Quiz You have $96 to spend on campground activities. You can rent a paddleboat for $8 per hour and a kayak for $6 per hour. Write an equation in standard form that models the possible hourly combinations of activities you can afford. List three possible combinations. ANSWER 8 p + 6 c = 96; 16 h kayak and 0 h paddleboat; 12 h paddleboat and 0 h kayak; 6 h paddleboat and 8 h kayak.