Lesson 6 — Solving Linear Equations in One Variable

Summary

Learners solve linear equations in one variable by keeping a balance: do the same thing to both sides, use inverse operations to isolate the variable, and check by substituting the solution back. They move from two-step equations to equations with the variable on both sides.

Objectives

• Solve linear equations in one variable, including multi-step equations and equations with the variable on both sides, and check the solution. (D05.S2.08.02)

Connection

A market stall sells baskets at a price you cannot see, plus a fixed delivery fee of 3 coins. You paid 11 coins for two baskets. What does one basket cost? The equation 2x + 3 = 11 says it: undo the +3 (subtract 3 from both sides), then undo the ×2 (divide both sides by 2) — x = 4. Solving an equation is untying a knot one careful move at a time, and it is how a buyer, a cook, or a builder finds a fair unknown.

Materials

• Balance-pan sheet
• Equation cards

Preparation

• Copy the balance-pan sheet and equation cards.
• Have worked examples ready: 2x + 3 = 11 → subtract 3 → 2x = 8 → divide 2 → x = 4; and 3x = x + 8 → subtract x → 2x = 8 → x = 4.
• Recall from Grade 7: two-step equations (D05.S2.07.01); from Lesson 5: the variable is the input.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) an equation is a balance — whatever you do to one side you must do to the other; (2) to isolate the variable, undo operations with their inverses in reverse order (add/subtract first, then multiply/divide); (3) a solution is checked by substituting it back. Because solving equations is a foundational skill, use explicit instruction, worked examples, and guided practice before independent work (Kirschner, Sweller & Clark, 2006). Watch for the slips of doing an operation to only one side and of forgetting to distribute (4(x − 2) = 4x − 8). The critical-thinking lens: “check your answer” is not a ritual — it is the honest test of whether the solution is right. The egalitarian lens: the same two moves solve the equation no matter who holds it — mathematics is a leveler, not a gate (philosophy §4). The global lens: the word “algebra” comes from al-jabr (“completion”) in al-Khwarizmi’s 9th-century treatise, where moving a term to the other side was described as completing or balancing — the same balance we use today (S-293). Retrieval: y = kx (Grade 7) and f(x) (Lesson 5) are related — solving an equation finds the input that makes two expressions equal.

Procedure

1. Gather (5 min). You can describe how one thing depends on another. Today you find the unknown inside an equation.
2. Meet the balance (10 min). An equation is a balance: both sides are equal, and they stay equal only if you do the same thing to both sides.
3. Worked example — two steps (10 min). Solve 2x + 3 = 11. Subtract 3 from both sides: 2x = 8. Divide both sides by 2: x = 4. Check: 2(4) + 3 = 11. ✓
4. Worked example — variable on both sides (10 min). Solve 3x = x + 8. Subtract x from both sides: 2x = 8. Divide by 2: x = 4. Check: 3(4) = 12 and 4 + 8 = 12. ✓
5. Practice with a partner (10 min). Solve the equation cards, writing each step and substituting to check. Trade and check each other’s work.
6. Close (5 min). Share one equation and how you untied it. Remember: do the same to both sides, undo in reverse order, and always check.

Differentiation

• Support: Start with one-step equations (x + 3 = 11) and a drawn balance; describe each move aloud for learners with low vision; model each inverse explicitly.
• Accessibility: Model the balance tactilely with counters or stones on a real beam and narrate each move; for learners with dyscalculia, keep a worked-step card visible and offload the arithmetic to a partner so the focus stays on doing the same to both sides.
• Extension: Solve 2x + 3 = 2x + 5 (no solution) and 2x + 3 = 2x + 3 (every number works), and explain what each ending means.

Assessment

• Formative (observation/performance): Can the learner solve a linear equation in one variable, showing the same operation on both sides, and verify the solution by substitution?
• Self-check: The learner asks, “Did I do the same thing to both sides every step? Did I undo add/subtract before multiply/divide? Did I substitute my answer back to check?”

Home connection

At home, set up a small unknown (a price, a number of items) as an equation, solve it, and check by plugging the answer back in — then explain the balance to someone.

Resources

• Solving linear equations is standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019) (S-026). The word “algebra” derives from al-jabr in al-Khwarizmi’s 9th-century treatise (MacTutor, S-293). On explicit instruction and worked examples for novice skills: Kirschner, Sweller & Clark (2006), S-011.