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Core Focus on Linear Equations

Learn how to simplify and solve multi-step equations by balancing variables and applying inverse operations. Practice examples and step-by-step solutions provided for a comprehensive understanding.

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Core Focus on Linear Equations

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  1. Lesson 1.6 Core Focus onLinear Equations Solving Multi-Step Equations

  2. Warm-Up Solve each equation. • 30 = 4x + 10 • 0.7x – 2.1 = 0 x = 5 x = −27 x = 3

  3. Lesson 1.6 Solving Multi-Step Equations Simplify and solve multi-step equations.

  4. Solving Multi-Step Equations • Simplify each side of the equation by distributing and combining like terms, when necessary. • If variables are on both sides of the equation, balance the equation by moving one variable term to the opposite side of the equals sign using inverse operations. • Follow the process for solving one- and two-step equations to get the variable by itself.

  5. Explore! Multi-Step Equation Step 1 Write the equation that is represented by the equation mat shown below. Step 2 There are variable cubes on both sides of the mat. Remove the same number of cubes from each side so that only one side has variable cubes remaining. Draw a picture of what is on the mat now. Step 3 What is the next step you must take to balance the equation mat? Remember that your goal is to isolate the variable. Draw a picture of what is on the mat now.

  6. Explore! Multi-Step Equation Step 4 Once the variables are isolated, x can be determined by dividing the integer chips on the opposite side equally between the remaining variable cubes. How many integer chips belong to each variable cube? What does this represent? Step 5 Draw a representation of 5x + 7 = 2x + 4 on an equation mat. Step 6 Repeat Steps 2–4. What is the solution to this equation?

  7. Example 1 Solve the equation for x. 4(2x – 7) = 20 Distribute. Add 28 to both sides of the equation. Divide both sides of the equation by 8.  Check the solution. 4(2x – 7) = 20 8x – 28 = 20 + 28 +28 8x = 48 8 8a x = 6a 4(2 ∙ 6 – 7) = 20 4(12 – 7) = 20 4(5) = 20 20 = 20

  8. Extra Example 1 Solve the equation for x. 2(5x + 3) = 26 x = 2

  9. Example 2 Solve the equation for x. – 2x + 9 = 4x – 15 There are no parentheses so there is no need to use the Distributive Property. Add 2x to both sides of the equation. Add 15 to both sides of the equation. Divide both sides of the equation by 6. Check the solution. – 2x + 9 = 4x – 15 + 2x + 2x 9 = 6x – 15 + 15 + 15 24 = 6x 6 6a 4 = xa –2(4) + 9 = 4(4) – 15 –8 + 9 = 16 – 15 1 = 1

  10. Extra Example 2 Solve the equation for x.7x + 42 = 2x + 12 x = −6

  11. Example 3 Solve the equation for x. 5(x + 8) = – 2(x – 13) Use the Distributive Property to remove parentheses. Add 2x to both sides of the equation. Subtract 40 from both sides of the equation. Divide both sides of the equation by 7.  Check the solution. 5(x + 8) = – 2(x – 13) 5x + 40 = – 2x + 26 +2x + 2x a 7x + 40 = 26 – 40 –40 7x = –14 7 7a x = –2 5(– 2 + 8) = – 2(– 2 – 13) 5(6) = – 2(– 15) 30 = 30

  12. Extra Example 3 Solve the equation for x.−3(x + 2) = x − 10 x = 1

  13. Example 4 Katie opened a coffee cart to earn some extra money. Her one-time equipment start-up cost was $460. It costs her $1 to make each cup of coffee. She plans to sell the cups of coffee for $3. How many cups will she need to sell before she breaks even? Let x represent the number of cups of coffee sold. Write an equation that represents the situation. 460 + 1x = 3x Amount charged per cup Start-up cost Cost per cup

  14. Example 4 Continued… Katie opened a coffee cart to earn some extra money. Her one-time equipment start-up cost was $460. It costs her $1 to make each cup of coffee. She plans to sell the cups of coffee for $3. How many cups will she need to sell before she breaks even? Subtract 1x from both sides of the equation. Divide both sides by 2.  Check the solution. Katie must sell 230 cups of coffee to break even. 460 + 1x = 3x – 1x –1x 460 = 2x 2 2a 230 = xa 460 + 1(230) = 3(230) 460 + 230 = 690 690 = 690

  15. Extra Example 4 Gary opened a booth at the farmer’s market selling ceramic mugs. His start-up costs were $280. It costs him $0.50 to make each mug. He sells each mug for $4.50. How many mugs will he need to sell before he breaks even? 70 mugs

  16. Communication Prompt Describe the process of solving a multi-step equation.

  17. Exit Problems Solve each equation. • 90 = 10(x + 7) • 6x − 13 = 4x + 11 • 2(x − 9) = 3(x − 5) x = 2 x = 12 x = −3

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