Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter goals

By the end of this chapter, you will be able to:

• explain what a variable is in plain English,

• tell the difference between an expression and an equation,

• use the balance idea to understand legal algebra moves,

• simplify basic expressions without treating algebra as magic.

Prerequisites (if this is hard, go back to Unit 0)

You need the Unit 0 ideas of equality, signed numbers, and basic arithmetic. This chapter begins algebra language.

Why this matters

Key idea

Many students think algebra begins when numbers disappear and letters invade the page.

That is not what is happening.

Algebra begins when we stop talking about one fixed number and start describing a pattern that can work for many numbers.

Real life: what it’s for

Algebra is what lets you say:

• “the total cost is base fee plus price per mile,”

• “the temperature after ℎ hours follows this rule,”

• “my salary after tax is this fraction of my original pay,”

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Chapter 1. What algebra actually is

• “my phone bill is fixed fee plus data usage.”

Without variables, every situation would have to be rebuilt from scratch.

A variable is a placeholder for a number

Key idea

A variable is a symbol that stands for a value.

If 𝑥 = 5, then

𝑥 + 2 = 5 + 2 = 7 .

If 𝑥 = −3, then

𝑥 + 2 = −3 + 2 = −1 .

The rule stays the same. The value changes because the input changed.

Note

Students sometimes hear “unknown” and imagine that a variable is a mysterious object. It is not mysterious. It is a placeholder. Once you accept that, the page becomes much calmer.

Expression vs equation

Key idea

An expression is a math phrase with no equal sign. Examples: 3 𝑥 + 1, 2 𝑎 − 5, 4 𝑦 + 6.

An equation is a statement that two expressions are equal. Examples: 3 𝑥 + 1 = 10, 2 𝑎 − 5 = 7.

So:

• expressions are simplified,

• equations are solved.

Worked example (no skipped steps)

Simplify the expression:

3 𝑥 + 2 𝑥

Step 1: identify like terms. Both terms involve 𝑥.

Step 2: add the coefficients.

3 + 2 = 5

Step 3: keep the variable part.

3 𝑥 + 2 𝑥 = 5 𝑥

Answer: 5 𝑥.

The balance idea

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Worked example (no skipped steps)

Decide whether

4 𝑚 − 7 = 9

is an expression or an equation.

There is an equal sign, so this is an equation. That means it can be solved.

The balance idea

Recipe (steps that always work)

To keep an equation true, do the same legal move to both sides.

If you add 4 to the left side, add 4 to the right side. If you divide the left side by 3, divide the right side by 3.

Worked example (no skipped steps)

Solve:

𝑥 + 3 = 10

Step 1: undo the +3 by subtracting 3.

Step 2: subtract 3 from both sides.

𝑥 + 3 − 3 = 10 − 3

Step 3: simplify.

𝑥 = 7

Check:

7 + 3 = 10

The equation is true, so the solution is correct.

Worked example (no skipped steps)

Solve:

2 𝑝 = 18

The variable is being multiplied by 2. Undo that by dividing both sides by 2: 2 𝑝

18

=

2

2

𝑝 = 9

Check:

2(9) = 18

Correct.

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Chapter 1. What algebra actually is

Bridge (why this matters later)

This chapter is structural, not cosmetic. Functions, trigonometry, calculus, and statistics all assume you can read symbols calmly and make legal algebra moves without drama.

Common mistakes (and how to avoid them)

• thinking a variable is a mysterious object instead of a placeholder,

• mixing up simplifying an expression with solving an equation,

• changing only one side of an equation,

• combining unlike terms such as 3 𝑥 + 2 into 5 𝑥.

Recap: what you’re taking away

Algebra is not a secret language. It is arithmetic plus symbols.

If you can tell the difference between an expression and an equation, and if you understand the balance idea, you already have the core mental model that the rest of the book will use again and again.

Problems

1.1. If 𝑥 = 4, evaluate 𝑥 + 3.

1.2. If 𝑥 = −2, evaluate 2 𝑥 + 1.

1.3. Simplify: 5 𝑎 + 3 𝑎.

1.4. Simplify: 7 𝑦 − 𝑦.

1.5. Decide whether 3 𝑥 + 2 is an expression or an equation.

1.6. Decide whether 3 𝑥 + 2 = 11 is an expression or an equation.

1.7. Solve: 𝑚 + 5 = 12.

1.8. Solve: 𝑝 − 4 = 9.

1.9. If 𝑎 = 3, evaluate 4 𝑎 − 7.

1.10. If 𝑏 = −5, evaluate 2 𝑏 + 6.

1.11. Simplify: 9 𝑚 − 2 𝑚 + 4.

1.12. Solve: 𝑞 − 8 = 6.

1.13. If 𝑥 = 6, evaluate 3 𝑥 − 4.

1.14. If 𝑡 = −3, evaluate 4 𝑡 + 2.

1.15. Simplify: 4 𝑝 + 𝑝 − 7.

1.16. Simplify: 9 𝑚 − 4 𝑚 + 2.

Problems

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1.17. Solve: 𝑛 − 7 = 5.

1.18. Solve: 3 𝑘 = 21.

Quick checklist (if you miss it, reread and try again)

• I can explain what a variable is without sounding mystical.

• I know the difference between an expression and an equation.

• I understand why legal moves in an equation must happen on both sides.

• I can solve a one-step equation and check it.

Chapter 2

Signed numbers without confusion