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📐 Math Basics · Lesson 3 / 10

Equations: A Balance for Finding Unknown Numbers

An equation is a balance with equal weight on both sides. One principle, that doing the same thing to both sides keeps the balance, lets you solve everything from moving terms to systems of equations and word problems.

⏱ About 20 min ✍️ 4 practice questions 🔁 Unlimited drills Updated 2026-10-08
🎯 By the end of this lesson you can
  • Use the properties of equality to explain why moving a term across the equals sign flips its sign
  • Solve and check linear equations that include parentheses and fractions
  • Solve a system of equations in two unknowns by elimination and by substitution
  • Turn a word problem into an equation by choosing what the unknowns stand for

1.An Equation Is a Balance

An equation is an equality that contains an unknown number. 2x + 3 = 11 means "2 times some number plus 3 is 11," and solving the equation means finding the x that makes it true.

Think of an equation as a balance with equal weight in both pans. Adding or subtracting the same number on both sides, or multiplying both sides by the same number or dividing them by the same number other than 0, does not tip the balance. These are the properties of equality, and solving an equation is the process of using them to leave x alone on one side.

2x + 3 = 11
2x + 3 − 3 = 11 − 3 → 2x = 8
2x ÷ 2 = 8 ÷ 2 → x = 4

2.Moving Terms: Why the Sign Flips

In the solution above, instead of subtracting 3 from both sides, you can say for short, "move +3 to the right side and it becomes −3." This is called moving a term (transposing). The sign flips not because of a rule you memorize, but because it is the result of subtracting the same number from both sides.

Linear equations are usually solved in this order. Expand the parentheses; if there are fractions or decimals, multiply both sides by the same number to make everything whole numbers; move the x terms to the left and the constant terms to the right; simplify; then divide both sides by the coefficient of x. Finally, check by putting the answer back into the original equation.

  • Common mistake 1: Not flipping the sign when moving a term.
  • Common mistake 2: When clearing fractions, multiplying only some of the terms on one side. If you multiply by 6, the 10 on the right must also become 60.
  • Common mistake 3: Not multiplying a negative sign in front of parentheses into the terms inside. −(x − 3) = −x + 3.
ExampleSolve 3(x − 2) = x + 8.
  1. Step 1: Expand the parentheses: 3x − 6 = x + 8.
  2. Step 2: Move terms: x goes to the left and −6 goes to the right. 3x − x = 8 + 6.
  3. Step 3: Simplify: 2x = 14.
  4. Step 4: Divide both sides by 2: x = 7.
  5. Check: Left side 3(7 − 2) = 3 × 5 = 15, right side 7 + 8 = 15. Both sides are equal, so it is correct.
Answerx = 7
ExampleSolve x/2 + x/3 = 10.
  1. Step 1: Multiply both sides by 6, the least common multiple of the denominators 2 and 3. Every term must be multiplied: 6 × x/2 + 6 × x/3 = 6 × 10.
  2. Step 2: Simplify: 3x + 2x = 60, so 5x = 60.
  3. Step 3: x = 60 ÷ 5 = 12.
  4. Check: 12/2 + 12/3 = 6 + 4 = 10, which is correct.
Answerx = 12

3.Checking Is Not Optional

A big advantage of equations is that you can verify the answer yourself. Put the value you found into the original equation (the very first one, not one from the middle of your work) and see whether both sides come out equal. If you plug it into an intermediate equation, you won't catch mistakes made before that step.

The same goes when AI solves an equation for you. However convincing the solution looks, plugging the answer into the original equation takes less than 1 minute. Checking is the surest way to decide whether to trust an AI's answer.

4.Solving a Formula for the Letter You Want

The properties of equality work not only for equations with numbers but also for formulas. A formula is an equation expressing the relationship among several letters, so you do the same operations to both sides until only the letter you want is left on one side. This is called rearranging a formula.

For example, the formula for converting a Celsius temperature C to a Fahrenheit temperature F is F = 1.8C + 32. If you know the Fahrenheit temperature and want the Celsius one, solve for C. Subtracting 32 from both sides gives F − 32 = 1.8C, and dividing both sides by 1.8 gives C = (F − 32) ÷ 1.8. The order is the same as for a linear equation: move the added number first, and divide by the multiplied number afterward. If you reverse the order, dividing F by 1.8 first and then subtracting 32, the 32 never gets divided by 1.8, and the formula comes out wrong.

F = 1.8C + 32
F − 32 = 1.8C
C = (F − 32) ÷ 1.8
ExampleWhat is 86 degrees Fahrenheit in Celsius?
  1. Step 1: Use the formula solved for C: C = (F − 32) ÷ 1.8.
  2. Step 2: Plug in F = 86: 86 − 32 = 54.
  3. Step 3: 54 ÷ 1.8 = 30.
  4. Check: Putting it into the original formula, 1.8 × 30 + 32 = 54 + 32 = 86, which is correct.
Answer30 degrees Celsius

5.Systems of Equations: When There Are Two Unknowns

With two unknowns, you need two equations. A system of equations asks for the x and y that satisfy both equations at once. The basic strategy is to eliminate one unknown and reduce it to a single linear equation.

Elimination adds or subtracts the two equations to get rid of one unknown, and substitution rewrites one equation in the form x = … or y = … and plugs it into the other. Either way, you get the same answer.

ExampleSolve x + y = 10 and x − y = 4 by elimination.
  1. Step 1: Adding the two equations makes y disappear: (x + y) + (x − y) = 10 + 4, so 2x = 14.
  2. Step 2: x = 7.
  3. Step 3: Plug it into the first equation: 7 + y = 10, so y = 3.
  4. Check: 7 + 3 = 10 and 7 − 3 = 4. Both equations hold.
Answerx = 7, y = 3
ExampleSolve y = 2x and x + y = 15 by substitution.
  1. Step 1: Put 2x from the first equation in place of y in the second: x + 2x = 15.
  2. Step 2: 3x = 15, so x = 5.
  3. Step 3: y = 2 × 5 = 10.
  4. Check: 10 = 2 × 5, and 5 + 10 = 15.
Answerx = 5, y = 10

6.Word Problems: Turning Words into Equations

Word problems are hard not because of the arithmetic but because of the translation. Having a fixed routine makes them much easier. First, let an unknown stand for what you want to find, and write down its unit. Second, find the relationship in the text that amounts to "is equal to" and set up an equation. Third, solve it. Fourth, put the answer back into the original wording and see whether it makes sense. If a count comes out negative or as a fraction, you most likely set up the equation wrong.

Example1 apple costs $1,200 and 1 pear costs $2,000. You bought 10 pieces of fruit in total and paid $15,200. How many apples and how many pears did you buy?
  1. Step 1: Unknowns: x apples and y pears.
  2. Step 2: Count relationship: x + y = 10. Cost relationship: 1,200x + 2,000y = 15,200.
  3. Step 3: Substitute: putting y = 10 − x into the second equation gives 1,200x + 2,000(10 − x) = 15,200, that is, 1,200x + 20,000 − 2,000x = 15,200.
  4. Step 4: Simplify: −800x = −4,800, so x = 6. Therefore y = 10 − 6 = 4.
  5. Check: 6 + 4 = 10 pieces, and 1,200 × 6 + 2,000 × 4 = 7,200 + 8,000 = $15,200, which matches the problem.
Answer6 apples and 4 pears
ExampleA father is 3 times as old as his son, and in 12 years he will be 2 times as old. How old is the son now?
  1. Step 1: Unknowns: the son is a years old now, and the father is 3a.
  2. Step 2: In 12 years: the father is 3a + 12 and the son is a + 12. The relationship is 3a + 12 = 2(a + 12).
  3. Step 3: 3a + 12 = 2a + 24 → 3a − 2a = 24 − 12 → a = 12.
  4. Check: Now the son is 12 and the father is 36, 3 times as old. In 12 years they will be 24 and 48, 2 times as old.
AnswerThe son is 12 (the father is 36)
When you hand a word problem to an AI, ask it to "start by telling me what you chose the unknowns to be." That makes it easy to check whether the setup step is right.

📌 Key points

  • Doing the same thing to both sides of an equation keeps it true
  • Moving a term flips its sign because it is the result of subtracting or adding the same number on both sides
  • If there are fractions, multiply every term by the least common multiple to get whole-number coefficients
  • Solve a system of equations by eliminating one unknown through elimination or substitution
  • Always check your answer in the original equation (for word problems, in the original wording)

✍️ Practice questions

Answer first, then open "Answer and explanation".

Q1. What is the solution of 5x − 7 = 2x + 8?

⭕ Correct

❌ Not quite — see the explanation

Answer and explanation
Answer ③ x = 5

5x − 2x = 8 + 7, so 3x = 15 and x = 5. Check: 5 × 5 − 7 = 18 and 2 × 5 + 8 = 18.

Q2. Solve the system 2x + y = 11, x − y = 1.

Answer and explanation
Answer x = 4, y = 3

Adding the two equations gives 3x = 12, so x = 4. From the second equation, 4 − y = 1, so y = 3. Check: 2 × 4 + 3 = 11 and 4 − 3 = 1.

Q3. Solving x + 3 = 10, someone wrote, "Move the 3 to the right side: x = 10 + 3 = 13." What went wrong?

⭕ Correct

❌ Not quite — see the explanation

Answer and explanation
Answer ① The sign was not flipped when the term was moved

Moving +3 makes it −3, so x = 10 − 3 = 7. A check shows 13 + 3 = 16 ≠ 10, so you can tell right away that it's wrong.

Q4. A phone plan charges a $10,000 base fee per month plus $2,000 for every 1GB of data. The bill came to $24,000. How many GB of data were used?

Answer and explanation
Answer 7GB

10,000 + 2,000x = 24,000 → 2,000x = 14,000 → x = 7. Check: 10,000 + 2,000 × 7 = $24,000.

🔁 Unlimited practice

Problems are generated endlessly. Type your answer and press "Check" to have it graded right away, or press "Show solution" to see a step-by-step solution in the same order as the lesson. You can choose the difficulty, and your streak of correct answers is counted.

  • Easy: ax + b = c
  • Medium: equations with x on both sides
  • Hard: equations with parentheses or fraction coefficients (including fraction answers)

These drills are generated in your browser with JavaScript, which is not running right now. Use the examples and practice questions above, then reopen this page with JavaScript turned on.

🤖 Try asking AI like this

Copy a prompt and replace the [ ] parts with your own situation. Don't take the answer on trust — check it against this lesson.

When you want to find where your solution went wrong

Here's how I solved the equation 3(x − 2) = x + 8: [my solution]. Don't tell me the answer first. Check line by line whether each line follows correctly from the one before. Just tell me which line is the first wrong one and why.

When practicing setting up word problems

Make 5 word problems with everyday situations that lead to a system of equations. For each one, I'll send you my unknowns and equations first, and you only judge whether the equations are right. Don't show the solutions or answers until I ask.

When checking an answer an AI gave you

Plug the solution you just found directly into the original equation and show me the value of each side separately. Use the original equation, not an intermediate one.
References
  • Standard middle school math textbook content (linear equations, systems of linear equations)

Reached every goal above? Mark the lesson complete.

📐 Math Basics

  1. 1Numbers and Operations: Fractions and Decimals Revisited
  2. 2Ratios and Rates: Reading Percentages Correctly
  3. 3Equations: A Balance for Finding Unknown Numbers
  4. 4Functions and Graphs: An Eye for Change
  5. 5Exponents and Logarithms: A World That Grows by Multiplying
  6. 6Geometry Basics: Area and Pythagoras
  7. 7Probability: Putting Numbers on Uncertainty
  8. 8Statistics: Mean, Median, and Variance
  9. 9Reading Data: The Traps in Graphs
  10. 10The Math for Understanding AI
📚 Worth reading
🧠What Generative AI Does and Where It Fails→ ✍️How to Write a Good Prompt→ 🔍Checking AI Answers→ 📚Using AI for Study and Work Without Plagiarism→
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