➗ Lesson 3.2: Algebra Basics
"Algebra" sounds scary, but at its heart it's just arithmetic with a missing number — and finding that number by keeping an equation balanced. In this lesson you'll meet variables, solve one- and two-step equations, read the coordinate plane, and use the formula sheet the test gives you. Gently, step by step, algebra becomes something you can do.
📚 What You'll Learn
By the end of this lesson, you will be able to:
- Understand variables and write simple expressions
- Solve one-step and two-step equations by keeping both sides balanced
- Read the coordinate plane — points, axes, and the idea of slope
- Use a given formula by plugging in numbers
⏱️ Estimated Time: 60–75 minutes (go at your own pace — there's no clock on you)
🎯 Project: Solve five equations (one- and two-step), translate one word problem into an equation, and plot two points on a hand-drawn grid.
In This Lesson
Arithmetic With a Missing Number
You already do algebra in your head. If a coffee costs $3 and you handed over $5 and got $2 back, you just solved 3 + ? = 5. Algebra simply gives that missing number a name — usually a letter like x — and a tidy method for finding it. That's the whole idea: find the value of the letter that makes the equation true.
The one golden rule that makes all of it work: whatever you do to one side of an equation, do to the other side too. An equation is a balance scale; keep it balanced and you can safely peel away everything around the x until it stands alone. Master that, and equations stop being scary.
🧠 Mindset
If algebra defeated you in school, know this: it's often taught too fast, with too many symbols at once. Here we go slowly, one step at a time, always tying it back to arithmetic you know. You do not need to be "smart at algebra" — you need to apply one balanced step, then the next. Plenty of people who "hated algebra" pass this test. You can be one of them. 🌱
Variables & Expressions
A variable is a letter that stands for a number we don't know yet (or that can change). An expression is a math phrase with numbers, variables, and operations — like 2x + 5 — but no equals sign. An equation has an equals sign and says two things are equal — like 2x + 5 = 11.
A key skill is turning words into an expression. The same keyword ideas from last lesson apply:
| Words | Expression |
|---|---|
| 5 more than a number | x + 5 |
| 3 less than a number | x − 3 |
| twice a number | 2x |
| a number divided by 4 | x ÷ 4 |
| 7 more than twice a number | 2x + 7 |
💡 "2x" means "2 times x"
In algebra, a number written right next to a letter means multiply: 2x is 2 × x, and 5a is 5 × a. There's no × sign needed. So if x = 3, then 2x = 6.
Solving Equations
To solve an equation is to get the variable alone on one side. You do that with inverse (opposite) operations — addition undoes subtraction, multiplication undoes division — always doing the same to both sides.
🧮 One-step: solve x + 5 = 12
- The 5 is added to x, so undo it by subtracting 5 from both sides.
- x + 5 − 5 = 12 − 5
- x = 7
Check: 7 + 5 = 12 ✓
🧮 One-step: solve 4x = 20
- x is multiplied by 4, so undo it by dividing both sides by 4.
- 4x ÷ 4 = 20 ÷ 4
- x = 5
Check: 4 × 5 = 20 ✓
🧮 Two-step: solve 2x + 3 = 11
- Undo the +3 first: subtract 3 from both sides → 2x = 8.
- Undo the × 2: divide both sides by 2 → x = 4.
Check: 2 × 4 + 3 = 8 + 3 = 11 ✓. Order tip: undo addition/subtraction first, then multiplication/division — the reverse of PEMDAS.
✅ Always check by plugging back in
Put your answer back into the original equation. If both sides match, you're right — a free, foolproof way to catch mistakes. On a multiple-choice test you can even test the answer choices: plug each into the equation and see which one works.
Word Problems into Equations
Algebra shines on word problems: name the unknown x, translate the sentence into an equation, and solve.
🧮 Worked Example: "A number tripled, then increased by 4, equals 19. Find the number."
- Let the number be x. "Tripled" = 3x; "increased by 4" = + 4; "equals 19" = = 19.
- Equation: 3x + 4 = 19.
- Subtract 4: 3x = 15. Divide by 3: x = 5.
The number is 5. Check: 3 × 5 + 4 = 15 + 4 = 19 ✓
Five more than twice a number is 17. What is the number?
- 5
- 6
- 11
- 12
✅ See the reasoning
"Twice a number" = 2x; "five more" = + 5; "is 17" = = 17. So 2x + 5 = 17. Subtract 5: 2x = 12. Divide by 2: x = 6. Check: 2 × 6 + 5 = 17 ✓. (You could also test the choices — only 6 works.)
The Coordinate Plane & Formulas
The coordinate plane is a grid made by two number lines: the horizontal x-axis and the vertical y-axis, crossing at the origin (0, 0). A point is named by an ordered pair (x, y) — always x first (across), then y (up/down).
| Point | How to plot it |
|---|---|
| (3, 2) | From the origin, right 3, then up 2 |
| (−4, 1) | Left 4, then up 1 |
| (0, −3) | Stay on the y-axis, down 3 |
Lines on the plane have a slope — how steep they are — found as rise over run (how much up, divided by how much across). If a line rises 2 for every 1 it runs right, its slope is 2 ÷ 1 = 2. You don't need deep slope skills to start; recognizing the idea and reading points off a graph goes a long way.
✅ You get a formula sheet — use it
The math test provides a formula sheet (area, perimeter, slope, and more), so you do not memorize them all. The skill is plugging numbers into the right formula. Example — distance = rate × time: a car at 50 mph for 4 hours travels 50 × 4 = 200 miles. Find the formula, substitute, and calculate.
Practice & Project
Solve for x: 3x − 5 = 16
- 5
- 6
- 7
- 21
✅ See the reasoning
Undo the −5 first: add 5 to both sides → 3x = 21. Undo the × 3: divide by 3 → x = 7. Check: 3 × 7 − 5 = 21 − 5 = 16 ✓. (Trap: 21 is 3x, not x — finish the last step.)
To plot the point (−2, 3), from the origin you go —
- right 2, then down 3
- left 2, then up 3
- up 2, then left 3
- right 3, then up 2
✅ See the reasoning
The pair is (x, y) = (−2, 3): x first — negative 2 means left 2 — then y — positive 3 means up 3. Always read x (across) before y (up/down), and remember negative x is left, negative y is down.
🎯 Your Project: Solve & Plot
Do these with pen and paper, checking each answer by plugging back in.
- Solve: x + 8 = 20
- Solve: 3x = 27
- Solve: 2x − 1 = 9
- Translate & solve: "Seven less than a number is 12. Find the number."
- Plot: draw a simple grid and mark the points (2, 3) and (−1, 2).
✅ Answers (check after you try)
1) Subtract 8: x = 12. 2) Divide by 3: x = 9. 3) Add 1 → 2x = 10, divide by 2 → x = 5. 4) x − 7 = 12 → add 7 → x = 19. 5) (2, 3) = right 2, up 3; (−1, 2) = left 1, up 2.
✅ Project Completion Checklist
- ☐ I solved the one-step equations
- ☐ I solved the two-step equation (subtract/add first, then divide)
- ☐ I turned a sentence into an equation and solved it
- ☐ I plotted two points, x-first
- ☐ I checked answers by plugging back in
👥 Working with a tutor or group?
Solve equations on a shared board, each person calling out the next balanced step ("subtract 3 from both sides"). Then play "build an equation": someone describes a number in words ("I tripled it and added 2 to get 20"), and the group writes and solves the equation. Saying the steps aloud, together, makes the balance-scale idea click.
🎯 Quick Quiz
Question 1: Solve: x − 4 = 9
Question 2: In the point (5, 2), which number is the x-coordinate (how far across)?
Tips & Common Mix-Ups
✅ Do's
- Do the same thing to both sides — that keeps the equation true.
- Undo addition/subtraction first, then multiplication/division, in two-step equations.
- Check by plugging back in, or test the answer choices.
❌ Common Mix-Ups
⚠️ Watch Out
- Stopping too early: "3x = 21" is not the final answer — divide to get x = 7.
- Doing to one side only: if you subtract 3 from the left, subtract 3 from the right too.
- Flipping x and y: a point is (x, y) — across first, then up/down.
- Negative signs: keep track of them carefully; a lost minus sign is a very common error.
✅ Affirmation
Every equation you solve — even slowly, even with a check every line — builds real algebra skill. You don't have to be fast or fearless; you have to balance one step at a time. That, you can absolutely do.
📓 Learning Journal
Keep your study journal going. After this lesson, take five minutes to write down:
- The golden rule in your own words (both sides balanced)
- What clicked for you
- What's still tricky (two-step? negatives? plotting?)
- A check method you'll use (plug back in / test choices)
- How you feel about algebra now vs. the start
✍️ This lesson's prompt: Write one real situation as an equation — "I had some money, spent $12, and had $8 left" becomes x − 12 = 8. Solve it. Seeing your own life as a solvable equation makes algebra feel natural.
📝 Lesson Summary
🎓 Key Takeaways
- A variable is a letter for an unknown; an expression has no equals sign, an equation does.
- Solve equations with inverse operations, doing the same to both sides; undo +/− first, then ×/÷.
- Points are (x, y) — across first, then up/down; slope is rise over run.
- A formula sheet is provided — the skill is plugging numbers into the right formula.
🎉 What You've Accomplished
You just did algebra — real algebra — and kept it balanced every step of the way. You can write expressions, solve one- and two-step equations, turn words into equations, read the coordinate plane, and use a formula. For a subject that scares so many people, that's a genuine breakthrough. Be proud of it. 🎉
❓ Common Questions at This Stage
Why undo addition before multiplication in two-step equations?
Because you're unwrapping the x from the outside in — the reverse of the order of operations. In 2x + 3, the +3 is the outer layer, so peel it off first, then divide by 2. Following that order keeps things simple.
What if there are negative numbers?
The same rules apply — just track the signs carefully. Adding a negative is like subtracting; two negatives multiplied make a positive. Go slowly with signs; they're the most common slip, and a check at the end catches them.
Do I need to graph lines and find equations of lines?
Some questions involve reading points and slope from a graph. Start by getting comfortable plotting points and understanding slope as rise over run. Deeper line work can come with more practice — but points and slope-reading handle a lot.
🔭 Looking Ahead
In Lesson 3.3, we finish math with geometry and data — area, perimeter, and volume (using that formula sheet), plus reading graphs and finding the mean, median, and probability. It's the last piece of the Math subject.
✅ Before the Next Lesson
- Do the five-problem project and check every answer by plugging back in.
- Write two real-life situations as equations and solve them.
- Write your Learning Journal entry.
🌟 Encouragement for the Journey
You just walked into the room labeled "Algebra" — the one so many people are afraid to enter — and found it's just balanced arithmetic, one step at a time. Every solved equation is proof you belong here. Keep balancing, keep checking, and keep that quiet confidence growing. See you in Lesson 3.3! 👋
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