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ANSWER

2. (. – 1, – 2), (2, 7 ). 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). ANSWER. y + 2 = 3( x + 1) or y – 7 = 3( x – 2).

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ANSWER

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  1. 2. ( –1, –2), (2, 7) 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) ory + 1 = –(x – 6) ANSWER y + 2 = 3(x + 1) ory– 7 = 3(x– 2) 3.A store rents 3 DVDs for $5, plus $3 for each additional DVD. Find the cost of renting 20 DVDs. ANSWER $56

  2. Review Homework

  3. Write an equation in point-slope form of the line that passes through (6, – 4) and has slope 2. 1. y + 4 = –2(x –6) ANSWER ANSWER y + 6 = 4(x + 1) or y –10 = 4(x–3) 2. Write an equation in point-slope form of the line that passes through (–1, –6) and (3,10). Daily Homework Quiz A travel company offers guided rafting trips for $875 for the first three days and $235 for each additional day. Write an equation that gives the total cost (in dollars) of a rafting trip as a function of the length of the trip. Find the cost for a 7-day trip. 3. C = 235t + 170, where Cis total cost and t is time (in days); $1815 ANSWER

  4. Methods to Represent Linear Functions • Slope Intercept Form: y = mx + b • Point-Slope Form: y – y1 = m(x – x1) • m = slope • (x1, y1) = point on the line • Standard Form: Ax + By = C • A, B, and C are real numbers. • Useful to model real life situations…. • Not useful for graphing

  5. EXAMPLE 1 Write equations in standard form Write these equations in standard form. y = 2x – 9 y = 6 - 5x y = 9 + x y + 1 = 3(x + 1) y – 2 = 5(x – 11) 2x – y = 9 5x + y = 6 x – y = -9 -3x + y = 2 -5x + y = -53

  6. 1 – (–2) m= –3 = = 1 –2 3 –1 Write an equation from a graph EXAMPLE 2 Write an equation in standard form of the line shown. SOLUTION STEP1 Calculate the slope. STEP2 Write an equation in point-slope form. Use (1, 1). y –y1=m(x –x1) Write point-slope form. y –1=– 3(x –1) Substitute 1 for y1,23 for m and 1 for x1.

  7. Write an equation from a graph EXAMPLE 2 STEP3 Rewrite the equation in standard form. 3x + y = 4 Simplify. Collect variable terms on one side, constants on the other.

  8. 2 Write an equation in standard form of the line through (3,–1) and (2, – 3). –3–(–1) m= 2 = = 2 –3 –2 –1 Write an equation from a graph EXAMPLE 2 for Examples 1 and 2 GUIDED PRACTICE SOLUTION STEP1 Calculate the slope. STEP2 Write an equation in point-slope form. Use (3, –1). y –y1=m(x –x1) Write point-slope form. y +1=2(x –3) Substitute 3 for x1, –1 for y1 and 2 for m.

  9. Write an equation from a graph EXAMPLE 2 for Examples 1 and 2 GUIDED PRACTICE STEP3 Rewrite the equation in standard form. – 2x + y = –7 Simplify. Collect variable terms on one side, constants on the other.

  10. Write an equation of the specified line. a. b. Blue line Red line a. 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. b. EXAMPLE 3 Write an equation of a line SOLUTION

  11. Find the missing coefficient in the equation of the line shown. Write the completed equation. Complete an equation in standard form EXAMPLE 4 EXAMPLE 3 EXAMPLE 4 SOLUTION STEP1 Find the value of A. Substitute the coordinates of the given point for xandyin the equation. Solve forA. Ax + 3y = 2 Write equation. A(–1) + 3(0) = 2 Substitute – 1 for x and 0 for y. Simplify. –A = 2 A = – 2 Divide by – 1.

  12. Complete an equation in standard form EXAMPLE 4 STEP2 Complete the equation. – 2x + 3y = 2 Substitute – 2 for A.

  13. for Examples 3 and 4 GUIDED PRACTICE Write equations of the horizontal and vertical lines that pass through the given point. 3. (–8, –9) SOLUTION STEP1 The y-coordinate of the given point is–9. This means that all points on the line have a y-coordinate of –9. An equation of the line is y=–9. STEP2 The x-coordinate of the given point is –8. This means that all points on the line have an x-coordinate of –8. An equation of the line is x= –8.

  14. for Examples 3 and 4 GUIDED PRACTICE Write an equation of the horizontal and vertical lines that pass through the given point. 4. (13, –5) SOLUTION STEP1 The y-coordinate of the given point is –5. This means that all points on the line have a y-coordinate of –5. An equation of the line is y=–5. STEP2 The x-coordinate of the given point is 13. This means that all points on the line have an x-coordinate of 13. An equation of the line is x= 13.

  15. Complete an equation in standard form EXAMPLE 4 EXAMPLE 3 for Examples 3 and 4 Write an equation of a line GUIDED PRACTICE Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation. 5. –4x+By = 7, (–1,1) SOLUTION STEP1 Find the value of B. Substitute the coordinates of the given point for xandyin the equation. Solve forB. –4x + By = 7 Write equation. –4(–1) + B(1) = 7 Substitute –1 for x and 1 for y. Simplify. B = 3

  16. Complete an equation in standard form EXAMPLE 4 for Examples 3 and 4 GUIDED PRACTICE STEP2 Complete the equation. – 4x + 3y = 7 Substitute 3 for B.

  17. Real Life Example • Standard Form: Ax + By = C • Example • You have $50 to spend at a used book store. • Paperbacks (x): $1, Hardcovers (y) $4 1x + 4y = 50 If I want to buy 7 hardcover books, how many paperback books could I buy? 1x + 4(7) = 50 -28 -28 x = 22

  18. Complete an equation in standard form EXAMPLE 4 EXAMPLE 3 for Examples 3 and 4 Write an equation of a line GUIDED PRACTICE Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation. 6.Ax+y = –3, (2, 11) SOLUTION STEP1 Find the value of A. Substitute the coordinates of the given point for xandyin the equation. Solve forA. Ax + y = –3 Write equation. A(2) +11 =–3 Substitute 2 for x and 11 for y. Simplify. 2A= –14 A= –7 Divide each side by 2.

  19. Complete an equation in standard form EXAMPLE 4 for Examples 3 and 4 GUIDED PRACTICE STEP2 Complete the equation. – 7x +y = –3 Substitute –7 for A.

  20. a. Write an equation in standard form that models the possible combinations of small vans and large vans that your class could fill. Solve a multi-step problem 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. b. Graph the equation from part (a). c. List several possible combinations.

  21. 8 s + 12 p l = Solve a multi-step problem EXAMPLE 5 SOLUTION a. Write a verbal model. Then write an equation. 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 8s + 12l=pto find the value of p. Substitute15forsand2forl. 8(15) + 12(2) = p 144 = p Simplify. Substitute 144 for pin the equation 8s+12l = p.

  22. ANSWER The equation 8s+12l=144 models the possible combinations. Solve a multi-step problem EXAMPLE 5 b. Find the intercepts of the graph. Substitute0fors. 8(0) + 12l = 144 l = 12 8s + 12(0) = 144 Substitute 0 for l. 8s + 12(0) = 144 s = 18

  23. 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. Solve a multi-step problem 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 sandlmake sense. 8s + 12(0) = 144

  24. Solve a multi-step problem EXAMPLE 5 for Example 5 GUIDED PRACTICE Solve a multi-step problem EXAMPLE 5 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.

  25. 8 s + 12 p l = Solve a multi-step problem EXAMPLE 5 for Example 5 GUIDED PRACTICE Solve a multi-step problem EXAMPLE 5 SOLUTION STEP1 Write a verbal model. Then write an equation. 8 students decide not to go on the class trip, so the class could fill 14 small vans and 2 large vans. Because your class could fill 14 small vans and 2 large vans, use (14, 2) as the s- and l-values to substitute in the equation 8s + 12l=pto find the value of p. Substitute14forsand2forl. 8(14) + 12(2) = p 136 = p Simplify. Substitute 136 for pin the equation 8s+12l = p.

  26. 4 l = 11 12 Solve a multi-step problem EXAMPLE 5 for Example 5 GUIDED PRACTICE ANSWER The equation 8s+12l=136 models the possible combinations. STEP2 Find the intercepts of the graph. Substitute0fors. 8(0) + 12l = 136 8s + 12(0) = 144 Substitute 0 for l. 8s + 12(0) = 136 s = 17

  27. 4 Plot the point(0,11 )and(17, 0).connect them with a line segment. For this problem only negative whole-number values of sandl make sense. 12 for Example 5 GUIDED PRACTICE STEP3 The graph passes through (17, 0),(14, 2), (11, 4),(8, 6), (5, 8) and (2, 10). So, several combinations are 17 small, 0 large;14 small 2 large; 11 small, 4 large;18 small, 6 large;5 small, 8 large;2 small, 10 large. 8s + 12(0) = 144

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