🍎Unlocking the Variable: Solving Multi-Step Linear Equations

Lesson plan

Objectives

• Apply the distributive property to simplify linear equations before solving.
• Identify and combine like terms on one or both sides of an equation.
• Solve multi-step linear equations using inverse operations to isolate a single variable.
• Verify solutions by substituting the calculated value back into the original equation.

Materials

• Dry-erase individual whiteboards and markers
• Equation tiles (physical or digital manipulatives)
• Printed 'Equation Mountain' graphic organizers
• Teacher computer with projector
• Calculators (for arithmetic verification only)
• Set of 'Find the Error' task cards

Warm-up

Display five equations on the board: 3x = 12, x + 5 = 11, 2x - 4 = 10, 1/2x = 8, and 15 - x = 5. Give students three minutes to solve these mentally or on their whiteboards. Once finished, ask students to turn to a partner and describe the 'inverse operation' they used for each. This activates prior knowledge of one and two-step equations before moving to multi-step logic.

Direct instruction

1. Step 1: Define a multi-step equation as any equation requiring three or more steps to isolate the variable, often starting with simplification.
2. Step 2: Introduce the 'D.C.M.A.M.' acronym (Distribute, Combine, Move variables, Add/Subtract, Multiply/Divide) as a roadmap for solving.
3. Step 3: Model Example 1: 3(x + 4) = 18. Demonstrate distributing the 3 first to get 3x + 12 = 18, then subtracting 12, then dividing by 3.
4. Step 4: Model Example 2: 5x + 2x - 4 = 10. Show students how to combine 5x and 2x first to get 7x - 4 = 10, emphasizing that we only combine terms on the same side of the equals sign.
5. Step 5: Model Example 3: 2(x - 5) + 4 = 14. This combines both distributive property and combining like terms before moving to inverse operations.
6. Step 6: Demonstrate the 'Check Your Work' phase by plugging the answer back into the original equation to see if both sides balance.
7. Step 7: Address the 'Equals Sign Wall'—draw a vertical line down from the equals sign to ensure students perform operations on both sides to maintain equality.

Guided practice

As a class, we will solve 4(2x - 3) + 5 = 21. Step 1: Distribute 4 into the parentheses (4*2x and 4*-3) to get 8x - 12 + 5 = 21. Step 2: Combine the constants -12 and +5 to get 8x - 7 = 21. Step 3: Add 7 to both sides, resulting in 8x = 28. Step 4: Divide both sides by 8, resulting in x = 3.5. We will then plug 3.5 back into the original equation to verify: 4(2(3.5)-3)+5 = 4(7-3)+5 = 4(4)+5 = 16+5 = 21.

Independent practice

Students will work through a tiered 'Equation Scavenger Hunt.' The classroom will have 12 cards posted. Each card has a linear equation and a letter. After solving the equation, they find their answer on a different card, which tells them which letter to write next on their answer sheet. This allows for self-correction: if they don't see their answer on any card, they know they made a mistake.

Closure

Review the D.C.M.A.M. steps one last time. For the exit ticket, students must solve the equation 2(x + 3) + x = 15 and write one sentence explaining the very first step they took to solve it.

Assessment

Mastery will be measured by the accuracy of the 10-problem worksheet and the 8-question quiz. A score of 80% or higher indicates the student can independently manage multi-step linear logic.

Differentiation

Scaffolding: Provide 'fill-in-the-blank' equation sheets where the steps are outlined but students provide the numbers. Use color-coded algebra tiles for visual learners. Extension: Challenge advanced learners with equations containing variables on both sides (e.g., 3x + 5 = 2x - 7) or equations resulting in 'No Solution' or 'Infinite Solutions'.

Mastering the Multi-Step: Equation Practice

Solve each linear equation below. You must show every step of your work, including the distributive property and combining like terms. Check your final answer by substituting it back into the original equation.

1. 2(x + 5) = 20
2. 3(y - 4) = 9
3. 5x + 2x - 10 = 11
4. -4(m + 2) = 12
5. 8k - 3k + 7 = 27
6. 2(3w - 1) = 10
7. 10 + 2(p - 3) = 20
8. 6 = 3(x - 2) + 3
9. 0.5(4x + 8) = 10
10. 15 = -3(x - 1)

Linear Equations Concept Quiz

1. What is the first step to solve 3(x + 4) = 21 using the distributive property?
   • Subtract 4 from 21
   • Divide 21 by 3
   • Multiply 3 by x and 3 by 4
   • Subtract 3 from both sides
   Answer: Multiply 3 by x and 3 by 4
2. Solve for x: 2x + 5x - 3 = 11
   • x = 2
   • x = 3
   • x = 1
   • x = 7
   Answer: x = 2
3. What is the inverse operation of subtraction?
   • Multiplication
   • Division
   • Addition
   • Square Root
   Answer: Addition
4. In the equation 4(x - 2) = 16, what is the value of x?
   • x = 4
   • x = 6
   • x = 8
   • x = 2
   Answer: x = 6
5. Which term is a 'like term' to 4x?
   • 4
   • x^2
   • -10x
   • xy
   Answer: -10x
6. Solve for n: -2(n + 5) = -20
   • n = 5
   • n = 10
   • n = -5
   • n = 15
   Answer: n = 5
7. If x = 3, which equation is true?
   • 2x + 1 = 5
   • 3x - 2 = 7
   • x + 10 = 12
   • 5x = 10
   Answer: 3x - 2 = 7
8. What happens when you multiply a number by its reciprocal?
   • The result is 0
   • The result is 1
   • The result is the number itself
   • The number becomes negative
   Answer: The result is 1

Equation Exploration: Home Edition

This assignment reinforces the multi-step solving techniques learned in class. Parents, please encourage your student to show the 'balance' of the equation by performing the same operation on both sides of the equals sign. Checking the answer at the end is a vital skill for building mathematical confidence.

• Solve the equation 5(x - 2) = 25 and show all work.
• Solve 3y + 4y - 5 = 9 and highlight the 'like terms' you combined.
• Explain in writing why 2(x + 3) is the same as 2x + 6.
• Complete the 'Error Analysis' problem: A student solved 2x + 4 = 10 and got x = 7. Find their mistake.
• Solve 10 = 2(w + 1) + 4 and check your answer through substitution.
• Create your own multi-step equation, solve it, and write a 'hint' for a classmate on how to start it.

Vocabulary

Variable · noun

A symbol, usually a letter, used to represent an unknown number.

"In the equation 2x = 10, 'x' is the variable."

Coefficient · noun

The numerical factor in a term containing a variable.

"In the term 5y, the number 5 is the coefficient."

Constant · noun

A number that does not change its value.

"In the expression 4x + 7, the number 7 is a constant."

Like Terms · noun

Terms that have the same variables raised to the same exponents.

"We can combine 3x and 5x because they are like terms."

Inverse Operation · noun

Operations that undo each other, such as addition and subtraction.

"Division is the inverse operation of multiplication."

Distributive Property · noun

Multiplying a single term by two or more terms inside a set of parentheses.

"Using the distributive property, 3(a + b) becomes 3a + 3b."

Equation · noun

A mathematical statement that two expressions are equal.

"Every equation must have an equals sign."

Isolate · verb

To get the variable by itself on one side of the equals sign.

"Our goal is to isolate 'x' to find its value."

Simplify · verb

To reduce an expression to its most basic form by combining terms.

"You should simplify the left side of the equation before solving."

Solution · noun

A value that makes an equation true when substituted for the variable.

"The solution to x + 2 = 5 is x = 3."

Activities

• Algebra Tile Race · 10 minutes

  Students use physical or digital algebra tiles to model multi-step equations. The teacher calls out a distributive property equation (e.g., 2(x+1)=6), and students must build it on their desks. The first group to correctly model and solve the equation by 'matching' tiles wins a point.

• Human Equation · 10 minutes

  Students receive large cards with numbers, variables, or operations. The teacher calls out an equation, and students must stand in order to form it at the front of the room. Another student acts as the 'Solver' and physically moves their classmates to show the balancing of the equation.

• Find the Flaw · 10 minutes

  Display a pre-solved multi-step equation that contains a common mistake (like failing to distribute to both terms). In pairs, students must troubleshoot the work, circle the error, and provide the correct calculation. This builds critical thinking and attention to detail.

• Equation Speed Dating · 15 minutes

  Students sit in two rows facing each other. Each student has a unique equation card. They have 2 minutes to trade cards, solve the new equation, and check each other's work. When the buzzer sounds, one row shifts down, and they repeat the process with a new partner.