1. ALGEBRA 1 Lesson 2-3 Warm-Up

2. ALGEBRA 1 Lesson 2-3 Warm-Up

3. ALGEBRA 1 “Equations With Variables on Both Sides” (2-3) (2-2) How can you solve an equation with variables on both sides? Tip: To solve an equation with variables on both sides, use the Addition or Subtraction Properties of Equality to get the variables on one side of the equal sign. Example: 5m + 3 = 3m – 9 -3m -3m Subtract 3m from both sides to get it on one side of the equal sign. 2m + 3 = - 9 - 3 - 3Subtract 3 from both sides to isolate 2m 2m = -12Divide both sides by 2 to isolate the m 2 2 1m = -6

4. ALGEBRA 1 Solve 5x – 3 = 2x + 12. 5x – 3= 2x + 12 – 2x –2xSubtract 2x from each side. 3x – 3= 12Combine like terms. + 3 + 3Add 3 to each side. 3x = 15Simplify. x=5Simplify. =Divide each side by 3. 3x33x3 15 3 Equations With Variables on Both Sides LESSON 2-3 Additional Examples

5. ALGEBRA 1 You can buy a skateboard for $60 from a friend and rent the safety equipment for $1.50 per hour. Or you can rent all items you need for $5.50 per hour. How many hours must you use your skateboard to justify buying your friend’s skateboard? Words: cost of plus equipment equals skateboard and friend’s rental equipment rental skateboard Define: let h = the number of hours you must skateboard. Equation: 60 + 1.5 h = 5.5 h Equations With Variables on Both Sides LESSON 2-3 Additional Examples

6. ALGEBRA 1 60 + 1.5h= 5.5h – 1.5h – 1.5h Subtract 1.5h from each side. 60=4hCombine like terms. You must use your skateboard for more than 15 hours to justify buying the skateboard. (continued) 15=hSimplify. 60 4 4h44h4 =Divide each side by 4. Equations With Variables on Both Sides LESSON 2-3 Additional Examples

7. ALGEBRA 1 “Equations With Variables on Both Sides” (2-3) (2-2) What happens when the variables on both sides of the equation cancel each other out? If the variable on both sides of an equation cancel each other out when you use the Addition or Subtraction Properties of Equality (Example: 2x = 2x), it will have an infinite number of solutions called an “identity” if both sides of the equation are equal (Example: 10 = 10) or no solution if both sides of the equation are not equal (Example: 10 = 7) Example: 3x + 9 = 3 + 3 (x + 2) 3x + 9 = 3 + 3 (x + 2) Distributive Property 3x + 9 = 3 + 3x + 3 2 3x + 9 = 3x + 9 Simplify -3x -3xSubtract 3x from both sides 9 = 9Identity (infinite number of solutions) Proof: The following function table shows that any solution for x works. x3x + 9=3 + 3(x + 2) True Statement? 1053(105) + 9 = 324 =3 + 3(105 + 2) = 3 + 3(107) = 3 + 321 =324 Yes 324 = 324 1,4203(1,420) + 9 = 4,269 =3 + 3(1,420 + 2) = 3 + 3(1,422) = 3 + 4,266 = 4,269 Yes 4,269 = 4,269 23,24 6 3(23,246) + 9 = 69,747 =3 + 3(23,246 + 2) = 3 + 3(23,248) = 3 + 69,744 = 69,747 Yes 69,747=69,747

8. ALGEBRA 1 “Equations With Variables on Both Sides” (2-3) (2-2) Example: 3x + 10 = 3 + 3(x + 2) 3x + 10 = 3 + 3(x + 2)Distributive Property 3x + 10 = 3 + 3 x + 3 2 3x + 10 = 3x + 9 Simplify -3x -3x Subtract 3x from both sides 10 ≠ 9no solutions Proof: The following function table shows that no solution work for x. x3x + 10=3 + 3(x + 2) True Statement? 123(12) + 10 = 46 =3 + 3(12 + 2) = 3 + 3(14) = 3 + 42 =45 No 46 ≠ 45 1053(105) + 10 = 325 =3 + 3(105 + 2) = 3 + 3(107) = 3 + 321 =324 No 325 ≠324 1,4203(1,420) + 10 = 4,270 =3 + 3(1,420 + 2) = 3 + 3(1,422) = 3 + 4,266 = 4,269 No 4,270 ≠ 4,269

9. ALGEBRA 1 Solve each equation. b.–6z + 8=z + 10 – 7z –6z + 8=z + 10 – 7z –6z + 8=–6z + 10Combine like terms. –6z + 8 + 6z=–6z + 10 + 6zAdd 6z to each side. 8≠10Not true for any value of z! This equation has no solution. a.4 – 4y=–2(2y – 2) The equation is true for every value of y, so the equation is an identity. 4 – 4y=–2(2y – 2) 4 – 4y=–2 2y – (-2) (2)Use the Distributive Property. 4 – 4y + 4y=–4y + 4 + 4yAdd 4y to each side. 4=4Always true! Equations With Variables on Both Sides LESSON 2-3 Additional Examples

10. ALGEBRA 1 Solve each equation. 1.3 – 2t = 7t + 42. 4n = 2(n + 1) + 3(n – 1) 3.3(1 – 2x) = 4 – 6x 4.You work for a delivery service. With Plan A, you can earn $5 per hour plus $.75 per delivery. With Plan B, you can earn $7 per hour plus $.25 per delivery. How many deliveries must you make per hour with Plan A to earn as much as with Plan B? 4 deliveries 1 no solution – 1919 Equations With Variables on Both Sides LESSON 2-3 Lesson Quiz