Inverse Equation Solving

Overview

This lesson introduces solving simple linear equations using inverse operations. Students will connect the algebra steps to what the graph or number line would show, and they will verify solutions by substitution.

Learning intentions

Students will be able to:

• identify what inverse operation undoes an operation in an equation
• solve linear equations of the form ax + b = c using inverse operations
• check their solution by substitution
• explain why the solution makes the equation true

Success criteria

• I can choose the correct inverse operation to isolate the variable.
• I can solve linear equations with integer coefficients and constants.
• I can substitute my answer back into the original equation and confirm it balances.
• I can describe each step as “doing the same thing to both sides”.

Curriculum links

• Algebra — solve linear equations and one-variable inequalities using graphical and algebraic techniques, and verify solutions by substitution.
• Algebra — graph linear relations on the Cartesian plane using digital tools where appropriate (as a quick link to how solutions appear).
• Algebra — create, expand, factorise, rearrange and simplify linear expressions using properties (as needed when simplifying expressions during solving).

Lesson structure (30 minutes)

1. 0–3 min · Retrieval warm-up. Teacher writes two quick “undoing” prompts (e.g. “If you add 7 then undo it: ___”; “If you multiply by -3 then undo it: ___”). Students answer in notebooks and share briefly.

2. 3–9 min · Direct teach: inverse operations. Teacher models solving x + 5 = 12 by doing the same inverse operation to both sides (subtract 5 from both sides), then checks by substitution. Students follow along on a worked example and underline the inverse operation used.

3. 9–13 min · Guided practice (think–pair–share). Teacher displays x − 3 = 8 and asks: “What undo operation goes with ‘minus 3’?” Students work in pairs to propose the inverse and perform the steps; teacher circulates and listens for “same thing to both sides”.

4. 13–18 min · Independent practice with quick checks. Students solve two equations independently:

• 2x = 10
• x + 4 = 9 After each one, students immediately verify by substitution (e.g. replace x with the answer and confirm both sides match).

5. 18–24 min · Connect to graphs (short, focused). Teacher shows that a solution is where both sides equal the same y-value for the same x. Using graphing software or a hand-drawn Cartesian plane, the teacher graphs y = x + 4 and y = 9 (a horizontal line), and points to their intersection as the solution for x + 4 = 9. Students match the intersection x-value to the algebraic solution.

6. 24–28 min · Whole-class misconception check. Teacher selects one common error example (e.g. solving x + 5 = 12 as x = 7 by subtracting incorrectly or changing only one side). Students vote: “Is it correct? What went wrong?” Teacher explains and restates the rule: same inverse operation on both sides.

7. 28–30 min · Exit ticket. Students complete one final equation and verification: x − 6 = 2 (solve, then substitute). Teacher collects for quick marking.

Resources

• Whiteboard and markers
• Student notebooks and pens
• Printed worksheet (1 page) with 3 equations: one worked space, two independent spaces, one exit ticket
• Optional: graphing software or a digital app that plots functions
• Graph paper or a pre-drawn Cartesian plane template
• Equation cards showing “Undo” prompts (add/subtract, multiply/divide) for teacher modelling

Assessment

• Formative checks during guided practice: teacher listens for correct inverse operation choice and “both sides” reasoning.
• Verification step review: teacher checks whether substitutions are performed and whether equality is confirmed.
• Exit ticket (x − 6 = 2): assessed for correct solution and correct substitution verification.

Differentiation

• Support: provide a step template (1) isolate variable, (2) inverse operation on both sides, (3) substitute and check. Include sentence starters such as “To undo +5, I subtract 5.”
• Support for language: highlight keywords in the task (undo, both sides, substitute, equal).
• Extension: offer a challenge equation for early finishers, such as 3x + 2 = 11 (students must use inverse operations carefully, possibly involving removing +2 then dividing by 3).
• SEN/EAL consideration: allow working on rough paper with a “box” for each step; encourage picturing the inverse as the reverse action.