Variables and Equations

Introduction

Algebra uses letters, called variables, to stand for numbers we do not know yet, or that can change. It lets us describe rules and solve puzzles in general. A phone plan costing 10 plus 2 per gigabyte is the rule cost = 10 + 2g, which works for any g.

Expressions

An expression combines numbers, variables and operations: 3x + 5. Terms with the same variable can be combined.

• 2x + 3x = 5x
• 4a + 7 - a + 2 = 3a + 9

Multiplication signs are often omitted: 3x means 3 times x. Brackets show that something is multiplied by everything inside: 2(x + 4) = 2x + 8.

Equations

An equation says two expressions are equal. Solving means finding the value of the variable that makes it true.

The key idea: whatever you do to one side, do to the other. This keeps the balance.

Example 1: 3x + 5 = 20

1. Subtract 5 from both sides: 3x = 15.
2. Divide both sides by 3: x = 5.

Check: 3(5) + 5 = 15 + 5 = 20. ✓

Example 2: 2(x - 4) = 18

1. Divide both sides by 2: x - 4 = 9.
2. Add 4: x = 13.

Check: 2(13 - 4) = 2 x 9 = 18. ✓

Variables on both sides

Example 3: 5x + 2 = 3x + 10

1. Subtract 3x from both sides: 2x + 2 = 10.
2. Subtract 2: 2x = 8.
3. Divide by 2: x = 4.

Check: left 5(4) + 2 = 22, right 3(4) + 10 = 22. ✓

Formulas

Formulas are equations you can rearrange. The distance formula d = s x t becomes t = d / s. To make a variable the subject, use inverse operations on both sides.

Word problems

1. Define the variable: "Let x be the number of tickets."
2. Write the equation from the story.
3. Solve.
4. Check against the words, not just the equation.

Example: a taxi charges 3 plus 2 per km. A fare was 17. How far? 3 + 2k = 17, so 2k = 14, so k = 7 km.

Common mistakes

• Forgetting to apply an operation to both sides.
• Sign errors when subtracting.
• Dividing only part of an expression. In 4x + 8 = 20, divide everything by 4 or subtract 8 first.
• Not checking the answer.

Try it yourself

Solve: 6x - 9 = 27, 3(x + 2) = 21 and 7x + 1 = 4x + 13. Check each by substitution. Write one equation for a real situation and solve it.

Summary

• Variables stand for unknown or changing numbers, and expressions can be simplified by combining like terms.
• Solve equations by doing the same inverse operation to both sides.
• Always check by substituting your answer back in.