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Chapter 1

Chapter 1. Section 3 Solving Equations. Solving Equations. 1. 5 x – 9 x + 3. 2. 2 y + 7 x + y – 1. x 3. y 3. 2y 3. 3. 10 h + 12 g – 8 h – 4 g. 4. + + – y. 5. ( x + y ) – ( x – y ). 6. –(3 – c ) – 4( c – 1). ALGEBRA 2 LESSON 1-3.

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Chapter 1

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  1. Chapter 1 Section 3 Solving Equations

  2. Solving Equations 1. 5x – 9x + 3 2. 2y + 7x + y – 1 x 3 y 3 2y 3 3. 10h + 12g – 8h – 4g 4. + + – y 5. (x + y) – (x – y) 6. –(3 – c) – 4(c – 1) ALGEBRA 2 LESSON 1-3 (For help, go to Lesson 1-2.) Simplify each expression.

  3. Solving Equations x 3 y 3 2y 3 x 3 y 3 2y 3 3y 3 x + y + 2y – 3y 3 x + 0y 3 x 3 x + (1 + 2 – 3)y 3 ALGEBRA 2 LESSON 1-3 1. 5x – 9x + 3 = (5 – 9)x + 3 = –4x + 3 2. 2y + 7x + y – 1 = 7x + (2 + 1)y – 1 = 7x + 3y – 1 3. 10h + 12g – 8h – 4g = (12 – 4)g + (10 – 8)h = 8g + 2h 4. + + – y = + + – = = = = 5. (x + y) – (x – y) = (x + y) + (y – x) = (x – x) + (y + y) = 0 + 2y = 2y 6. –(3 – c) – 4(c – 1) = (c – 3) – 4c – 4(–1) = c – 3 – 4c + 4 = (1 – 4)c + (–3 + 4) = –3c + 1 Solutions

  4. Let’s look at a problem. 5 = 5 +3 8 = 5 +3 8 = 8 What can be done to fix it, do not change the left side.

  5. b 7 = 7 - 2 7 = 5 What can be done to fix it, do not change the right side. - 2 5 = 5

  6. a 6 = 6 x 4 24 = 6 What can be done to fix it, do not change the left side. x 4 24 = 24

  7. 35 = 35 ÷ 5 35 = 7 What can be done to fix it, do not change the right side. ÷ 5 7 = 7

  8. a = a - b a - b = a What can be done to fix it, do not change the left side. - b a - b = a - b

  9. What was the point? Whatever you do to one side of an equation you MUSTdo to the other!

  10. Solving an Equation with a Variable on Both Sides Remember to move all variables to one side, constants to the other. 13y + 48 = 8y - 47 -8y-8y subtract 8y from each side 5y + 48 = - 47 - 48- 48 subtract 48 from each side 5y = -95 ÷5 ÷5 divide each side by 5 y = -19

  11. Using the Distributive Property Remember that everything in the parenthesis gets what is being multiplied on the outside. 3x – 7(2x – 13) = 3(-2x + 9) 3x – 14x + 91 = -6x + 27 -11x + 91 = -6x + 27 64 = 5x 12.8 = x

  12. Try These Problems Solve each equation. • 2(y – 3) + 6 = 70 • 6(t – 2) = 2(9 – 2t)

  13. Solving a Formula for One of its Variables Solve the formula for the area of a trapezoid for h. Multiply both sides by 2. Divide each side by b1+b2

  14. Solving an Equation for One of its Variables Solve for x and find any restrictions on a and b. Multiply by the LCD (ab) Gather like terms “un-distribute” the x Divide by (a-b) Now find restrictions What is the LCD of this equation?

  15. Restrictions on a and b What number can the denominator NOT be? Zero If the denominator is a-b, a-b ≠ 0 When stating restrictions we name only one variable on each side of the ≠ a – b ≠ 0 + b + b a ≠ b This means the restriction is: a ≠ b

  16. The Problem and Complete Answer THE COMPLETE ANSWER!!

  17. Try These Problems Solve for x. Find any restrictions a) ax + bx - 15 = 0 b)

  18. Homework • Practice 1.3 # 1 – 6, 10 – 17 • Word problems tomorrow

  19. 3 5 w = 13 Divide each side by 5. Relate: 2 • width + length = perimeter Define: Let w = the width. Then 3w = the length. Write: 2 w + 3w = 68 4 5 3w = 40 Find the length. 3 5 4 5 The width is 13 ft and the length is 40 ft. Solving Equations Adrian will use part of a garage wall as one of the long sides of a rectangular rabbit pen. He wants the pen to be 3 times as long as it is wide. He plans to use 68 ft of fencing. Find the dimensions of the pen. 5w = 68 Add. Check: Is the answer reasonable? Since the dimensions are about 14 ft by 41 ft and 14 + 14 + 41 = 69, the perimeter is about 69 ft. The answer is reasonable.

  20. Try this problem A rectangle is twice as long as it is wide. Its perimeter is 48 cm. Find its dimensions. 2w = l 48 = 2(2w) + 2w 48 = 4w + 2w 48 = 6w 8 = w So the length is 16 and the width is 8.

  21. Relate: Perimeter equals the sum of the lengths of the four sides. Define: Let x = the length of the shortest side. Then 2x = the length of the second side. Then 3x = the length of the third side. Then 6x = the length of the fourth side. Solving Equations The sides of a quadrilateral are in the ratio 1 : 2 : 3 : 6. The perimeter is 138 cm. Find the lengths of the sides. Write: 138 = x + 2x + 3x + 6x 138 = 12xCombine like terms. 11.5 = x 2x = 2(11.5) 3x = 3(11.5)6x = 6(11.5)Find the length of= 23 = 34.5 = 69 each side. Check:  Is the answer reasonable? Since 12 + 23 + 35 + 69 = 139, the answer is reasonable. The lengths of the sides are 11.5 cm, 23 cm, 34.5 cm, and 69 cm.

  22. Try This Problem The sides of a triangle are in the ratio 12:13:15. The perimeter is 120 cm. Find the lengths of the sides of the triangle. 12x + 13x + 15x = 120 40x = 120 x = 3 12(3) = 36 13(3) = 39 15(3) = 45 Check that the answer makes sense (36 + 39 + 45 = 120) The sides of the triangle measure 36 cm, 39 cm, and 45cm.

  23. Consecutive Number Problems • Remember that consecutive means in order:1,2,3,… so x, x+1,x+2 • Consecutive odds/evens means:x, x+2, x+4,…. • Usually the problems state that the numbers are integers….this means we don’t need to worry about decimals and fractions

  24. Solving Equations Find three consecutive integers that add to 90. Let x be the first integer. x + (x + 1) + (x + 2) = 90 3x + 3 = 90 3x = 87 x = 29 So the integers are 29, 30 and 31.

  25. Try This Problem The sum of four consecutive integers is 298. What are the numbers? x + (x+1) + (x +2) + (x + 3) = 298 4x + 6 = 298 4x = 292 x = 73 So the integers are 73, 74, 75, and 76.

  26. Rate, Time, Distance Problems • If the word problem involves transportation (cars, busses, planes, bicycles, canoes, etc…) it will probably involve rate, time and distance. • rate x time = distance (rt = d)

  27. 1225 6 50t = Solve for t. 1 12 t = 4 h or about 4 h 5 min Relate: distance first plane travels = distance second plane travels. Define: Let t = the time in hours for the second plane. Then t + = the time in hours for the first plane. 35 60 Write: 400t = 350 (t + ) 7 12 1225 6 400t = 350t + Distributive Property 2 3 Solving Equations A plane takes off from an airport and flies east at a speed of 350 mi/h. Thirty-five minutes later, a second plane takes off from the same airport and flies east at a higher altitude at a speed of 400 mi/h. How long does it take the second plane to overtake the first plane? Check:  Is the answer reasonable? In 4 h, the second plane travels 1600 mi. In 4 h, the first plane travels about 1600 mi. The answer is reasonable.

  28. Homework • Practice 1-3 # 7 – 9, 18 - 21

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