Chapter 08

Expressions, variables and equations

This is the chapter the course is named for. In arithmetic you are given all the numbers and you work out the answer. Here one of the numbers is missing, and a letter stands in its place until it is found.

Everything else stays the same, because the rules that apply to numbers apply to the letter standing in for one.

Expressions and equations

A variable is a letter that stands for a number. An expression is numbers, variables and operations with no equals sign. An equation has an equals sign in it.

  • Variables are letters like x, y and n.
  • Expressions look like x + 5, or 3y, or 2x + 7. There is nothing to solve.
  • Equations look like x + 5 = 12. There is something to solve.
  • Whatever you do to one side of an equation, do to the other side too.

Solving x + 5 = 12

Subtract 5 from both sides.

x = 12 − 5

x = 7

Key words

Much of pre-algebra is vocabulary. A question names what it wants in these terms, and the term decides which arithmetic answers it.

Variable
A letter that stands for a number.
Expression
Numbers, variables and operations with no equals sign, like 2x + 5.
Equation
A statement that two expressions are equal, like 2x + 5 = 11.
Coefficient
The number multiplied by a variable, like the 3 in 3x.
Constant
A number on its own, with no variable, like the 5 in 2x + 5.

Examples

Each one is set out the way a solution is written on paper: the question, the steps in order, then the answer.

Example 1. Putting a value in

Work out 8 × (x + 3) when x = 2.

Solution

  1. Put 2 where the x is: 8 × (2 + 3).
  2. Brackets first: 2 + 3 = 5.
  3. Now multiply: 8 × 5 = 40.

40

Example 2. Substitution

If x = 6, what is x + 4?

Solution

  1. Put 6 where the x is: 6 + 4.
  2. Add: 6 + 4 = 10.

10

Example 3. Substitution with a coefficient

If y = 4, what is 3y + 2?

Solution

  1. 3y means 3 times y, so put 4 in: 3(4) + 2.
  2. Multiply first: 12 + 2.
  3. Then add: 14.

14

Example 4. Evaluating an expression

If x = 3, what is 2x + 5?

Solution

  1. Put 3 in for x: 2(3) + 5.
  2. Multiply first: 6 + 5.
  3. Then add: 11.

11

Work an equation out

Two boards. The first solves an equation with one letter in it. The second puts a value in place of the letter and works the expression out. Both show every step, and both keep fractions exact, so a third is 1/3 here and not 0.333333333333, which is how you are expected to write it.

Solve an equation

One letter, one answer, and every step it took to get there.

Try

Evaluate an expression

Give the letter a value, and see the substitution before the answer.

Where this leads

The step-by-step solver
The same two boards on a page of their own, with what they accept and what they turn down set out in full.
Inequalities and one-step equations
The balance rule applied to equations that are solved in a single move.