Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
By the end of this chapter, you will be able to:
• solve one-step equations,
• solve two-step equations in the correct order,
• check whether a proposed solution is correct,
• understand equations as balance problems, not guesswork.
Prerequisites (if this is hard, go back to Unit 0)
You need the balance idea from Chapter 1 and inverse operations with signed numbers.
Why this matters
Key idea
Solving an equation means finding the value of the variable that makes the statement true.
The goal is not to “move things around.” The goal is to isolate the variable legally.
Real life: what it’s for
Equations answer real questions:
• “At what quantity do revenue and cost match?”
• “How many hours will it take to reach a target?”
• “What test score do I need on the final?”
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One-step equations
Recipe (steps that always work)
Undo the operation attached to the variable. Do the same thing to both sides.
Worked example (no skipped steps)
Solve:
𝑥 + 7 = 15
Subtract 7 from both sides:
𝑥 + 7 − 7 = 15 − 7
𝑥 = 8
Check:
8 + 7 = 15
Correct.
Worked example (no skipped steps)
Solve:
3 𝑥 = 21
Divide both sides by 3:
3 𝑥
21
=
3
3
𝑥 = 7
Check: 3(7) = 21.
Two-step equations
Key idea
If the variable is part of a two-step process, undo the steps in reverse order. This is the algebra version of taking off a jacket before a shirt.
Worked example (no skipped steps)
Solve:
2 𝑥 + 5 = 19
Step 1: undo the +5 first.
2 𝑥 + 5 − 5 = 19 − 5
2 𝑥 = 14
Chapter 6. Solving one-step and two-step equations
Step 2: divide by 2.
2 𝑥
14
=
2
2
𝑥 = 7
Check:
2(7) + 5 = 14 + 5 = 19
Correct.
Worked example (no skipped steps)
Solve:
𝑥 − 3 = 2
4
Step 1: add 3 to both sides.
𝑥 − 3 + 3 = 2 + 3
4
𝑥 = 5
4
Step 2: multiply both sides by 4.
𝑥
4 ·
= 4 · 5
4
𝑥 = 20
Common mistakes (and how to avoid them)
• doing different operations on the two sides,
• dividing before removing a constant,
• forgetting to check the answer,
• treating the equal sign as a place to restart instead of a balance statement.
Recap: what you’re taking away
Solving equations is mostly about staying calm and undoing operations in reverse order.
If you can keep the balance idea in view and check your answer at the end, one-step and two-step equations become routine instead of stressful.
Problems
6.1. Solve: 𝑥 + 9 = 14.
6.2. Solve: 𝑦 − 6 = 11.
6.3. Solve: 4 𝑚 = 28.
6.4. Solve: 𝑝 = 3.
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6.5. Solve: 2 𝑥 + 7 = 23.
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6.6. Solve: 3 𝑎 − 4 = 11.
6.7. Solve: 𝑡 + 2 = 6.
3
6.8. Solve: 5 − 2 𝑘 = 17.
6.9. Solve: 𝑥 − 11 = −4.
6.10. Solve: −5 𝑦 = 35.
6.11. Solve: 4 𝑚 − 9 = 19.
6.12. Solve: 𝑧 − 2 = 3.
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6.13. Solve: 𝑥 = 5.
4
6.14. Solve: 3 𝑦 − 2 = 16.
6.15. Solve: 2 𝑚 + 9 = 23.
6.16. Solve: 𝑛 + 4 = 9.
3
6.17. Solve: 5 − 𝑝 = 11.
6.18. Solve: −2 𝑥 = 14.
6.19. Solve: 7 𝑘 + 5 = 40.
6.20. Solve: 𝑎 − 3 = 1.
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Quick checklist (if you miss it, reread and try again)
• I can isolate the variable using inverse operations.
• I can undo a two-step process in reverse order.
• I know how to check a solution.
Multi-step equations that usually
make students freeze