Letters, expressions and equations

A letter holds the place of a number

You can write a rule before you know the number in it. A notebook costs an unknown number of dollars. Let n be the cost of one notebook in dollars. Three identical notebooks cost 3 × n, usually written 3n. With a $2 delivery charge, the total cost is 3n + 2 dollars.

The letter stands for a number, not for the object “notebook”. You could choose c instead, provided you say what c means and use it consistently. Every n in this rule represents the same cost. The notation 3n means multiplication, not a two-digit number beginning with 3.

An expression, such as 3n + 2, gives a rule for a value. It has no equals sign telling you what that value must be.

Check your understanding 1 Choose and use a letter

A ribbon has an unknown length in centimetres. A second ribbon is 6 cm longer. Choose a letter for the first length, say what it represents, and write an expression for the second length.

Your letter, its meaning and the second length

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One choice is: let r be the first ribbon’s length in centimetres. The second length is r + 6 cm. You might use a different letter; its meaning and its use must agree. The expression 6r would describe six times the first length, which is a different relationship.

Read the operation and its order

A letter can take the place of a number in each of the four operations:

• a + 3: add 3 to a.
• a-3: subtract 3 from a.
• 3a: multiply a by 3. You can also write a × 3.
• a/3: divide a by 3. You can also write a÷3.

Addition and multiplication allow the two numbers to swap: a + 3 = 3 + a and 3a = a × 3. Subtraction and division need more care. “Subtract a from 3” is 3-a, not a-3. “Divide 3 by a” is 3/a, not a/3. For a = 6, those two divisions give 1/2 and 2 respectively.

Check your understanding 2 Move between words and symbols

Write an expression for each description, then describe the last expression in words:

• subtract b from 9;
• share b litres equally among 4 containers;
• 7b.

Your two expressions and explanation of 7b

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The expressions are 9-b and b/4 litres per container. The expression 7b means seven times b. It does not mean 7 + b. Testing b = 8 helps distinguish the sharing expressions: b/4 gives 2 litres per container, whereas 4/b gives 1/2.

Combine quantities of the same kind

4a + 3a = 7a: four lots plus three lots make seven lots. Therefore 4a + 3a + 5 = 7a + 5. The five is a number of ones, not another five lots of a. For a = 2, the original and simplified expressions both give 19.

The number multiplying a letter is its coefficient. An a on its own means 1a. Keep each addition or subtraction with its term. For 9b-2b + 4, nine lots minus two lots leave seven lots, so the result is 7b + 4.

Different letters need separate groups. For example,

You cannot combine this to 8a or 8b: the two letters may represent different numbers. Fractions of the same unknown can combine too: a/3 + 2a/3 = a. The three thirds make one whole a.

Check your understanding 3 Repair a simplification

A learner writes 6p-2p + 3q + 5 = 12p. Simplify the original expression and explain why the learner’s grouping does not work.

Your simplified expression and reason

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6p-2p + 3q + 5 = 4p + 3q + 5. Only the two p terms can combine. Keep the q term and the constant separate, and subtract the two lots of p. As a check, with p = 2 and q = 1, the original and 4p + 3q + 5 both give 16; 12p gives 24.

Substitute a value for each letter

To evaluate an expression, replace each letter with its given number, then calculate. Multiply or divide before adding or subtracting.

For the notebook rule 3n + 2, if n = 4, the total is 3 × 4 + 2 = 14 dollars. It is not 3 × (4 + 2): the delivery charge is added once, after the three notebook costs.

For 2a + 3b, with a = 2 and b = 3, the value is 2 × 2 + 3 × 3 = 4 + 9 = 13. Give each letter its own stated value. For a/2 when a = 3, the value is 3/2 = 11/2; substitution need not give a whole number.

Check your understanding 4 Use both given values

Find the value of 5a-2b + 1 when a = 3 and b = 4. Write the substitution before calculating.

Your substitution and value

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5 × 3-2 × 4 + 1 = 15-8 + 1 = 8. The b term uses 4, not the value 3 given for a. The subtraction remains subtraction.

An equation makes a claim of equality

An equation says that two expressions are equal. Solving it means finding the number that makes the claim true. If the notebook total is $17, then 3n + 2 = 17.

Both sides can be pictured as equal loads on a balance. Removing the same load from each side preserves balance. Changing only one side would break equality. To isolate n:

• Subtract 2 from both sides: 3n = 15.
• Divide both sides by 3: n = 5.
• Check in the original equation: 3 × 5 + 2 = 17.

For 4x-3 = 17, add 3 to both sides to undo subtraction: 4x = 20. Divide both sides by 4 to get x = 5. Check 4 × 5-3 = 17. An operation is undone by its inverse, not by repeating it.

Sometimes the letter appears on both sides. For 5n + 2 = 2n + 23, subtract 2n from both sides to get 3n + 2 = 23. Subtract 2 to get 3n = 21, then divide by 3: n = 7. Checking gives 37 on each side. You could instead subtract 2 first; each step must keep the two sides equal.

Whole-number coefficients do not force a whole-number answer. For 6x = 9, dividing both sides by 6 gives x = 9/6 = 11/2. Checking gives 6 × 11/2 = 9.

Check your understanding 5 Explain the inverse steps

A number is multiplied by 5 and then 7 is subtracted. The result is 28. Write an equation and solve it. Show how your steps keep equality, then check your answer in the original equation.

Your equation, working and check, or work on paper

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5x-7 = 28. Add 7 to both sides: 5x = 35. Divide both sides by 5: x = 7. Check 5 × 7-7 = 28. Subtracting another 7 would not undo the original subtraction.

Change the unknown, keep the story

A taxi-fare model is 4 + 2d dollars for a journey of d km. Using this stated model, find the fare for 6 km, then the distance corresponding to a $20 fare. State the unit of each answer. This is a model for this task, not an actual company’s pricing.

Your fare, distance and working