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Explore: Letters for numbers

Algebra · step 2 of 5

A question to start

A phone plan costs 20 dollars a month plus 3 dollars for every gigabyte of data you use. What does a month cost if you use 4 gigabytes? 10? 25? You can work out each answer separately, but there is a better way: write one short rule that works for every number of gigabytes at once. To do that, you need a name for "the number of gigabytes", whatever it turns out to be. In algebra that name is a letter.

The idea

Variables

A variable is a letter that stands for a number. It is the bag of marbles from Play, with a label on it. Any letter will do; \(x\), \(n\) and \(t\) are popular. If \(g\) is the number of gigabytes, the phone bill is

\[20 + 3g \text{ dollars.}\]

Some ways algebra is written:

  • \(3g\) means \(3 \times g\). We leave out the times sign, because \(\times\) looks too much like \(x\).
  • \(\dfrac{n}{2}\) or \(n \div 2\) means \(n\) divided by 2.
  • \(n^2\) means \(n \times n\).
  • \(1g\) is just written \(g\).

Expressions

An expression is a mixture of numbers, letters and operations, such as \(20 + 3g\) or \(2n + 10\). It does not say that anything equals anything; it is like a phrase, not a sentence. The parts joined by \(+\) or \(-\) are called terms. In \(20 + 3g\) the terms are \(20\) and \(3g\). The number in front of a letter, here 3, is its coefficient.

Substitution: putting a number in

To find the value of an expression, substitute (put in) a number for the letter and do the arithmetic. For the phone bill with \(g = 4\):

\[20 + 3 \times 4 = 20 + 12 = 32 \text{ dollars.}\]

With \(g = 10\) it is \(20 + 30 = 50\) dollars. One rule, any number of gigabytes. Remember the order of operations: multiply before you add, unless brackets say otherwise.

Collecting like terms

Like terms have exactly the same letter part. \(3n\) and \(5n\) are like terms: three \(n\)'s and five \(n\)'s make eight \(n\)'s, so \(3n + 5n = 8n\). But \(3n\) and \(5\) are not like terms (one has a letter, one does not), and nor are \(n\) and \(n^2\). You can only add or subtract like terms, the way you can add apples to apples.

\[4a + 7 + 2a - 3 = 6a + 4\]

Equations

An equation says that two things are equal. It has an equals sign, and it is like a balanced scale: the left side weighs the same as the right side. \(x + 3 = 7\) is the marble puzzle from Play written in one line. Solving the equation means finding the value of \(x\) that makes it true.

The golden rule is the same as before: do the same thing to both sides.

\[ \begin{aligned} x + 3 &= 7\\ x + 3 - 3 &= 7 - 3 \qquad \text{(take 3 from both sides)}\\ x &= 4 \end{aligned} \]

For a two-step equation, undo the adding or subtracting first, then the multiplying or dividing, just like working backwards in Play:

\[ \begin{aligned} 2x + 1 &= 15\\ 2x &= 14 \qquad \text{(take 1 from both sides)}\\ x &= 7 \qquad \text{(divide both sides by 2)} \end{aligned} \]

The mind-reading trick, explained

Think of a number \(n\). Double it: \(2n\). Add 10: \(2n + 10\). Halve it: \(n + 5\). Take away the number you first thought of: \(n + 5 - n = 5\). The \(n\) disappears, which is why the answer is always 5. Algebra turns "it always seems to work" into "here is why it must work".

Worked examples

1. Writing an expression from words. A taxi charges a 4 dollar starting fee plus 2 dollars per kilometre. Write the cost for \(k\) kilometres, and find the cost of a 9 km ride.

  • Each kilometre costs 2 dollars, so \(k\) kilometres cost \(2k\) dollars. Add the fee: cost \(= 4 + 2k\).
  • For \(k = 9\): \(4 + 2 \times 9 = 4 + 18 = 22\) dollars.

2. Working backwards with an equation. The same taxi ride cost 30 dollars. How far was it?

  • Write the equation: \(4 + 2k = 30\).
  • Take 4 from both sides: \(2k = 26\).
  • Divide both sides by 2: \(k = 13\).
  • Check: \(4 + 2 \times 13 = 4 + 26 = 30\). The ride was 13 km.

3. Simplifying and then solving. Solve \(3x + 2x - 4 = 21\).

  • Collect like terms: \(3x + 2x = 5x\), so \(5x - 4 = 21\).
  • Add 4 to both sides: \(5x = 25\).
  • Divide both sides by 5: \(x = 5\).
  • Check: \(3 \times 5 + 2 \times 5 - 4 = 15 + 10 - 4 = 21\). Correct.

4. A perimeter puzzle. A rectangle is \(w\) cm wide and 3 cm longer than it is wide. Its perimeter (the distance all the way round) is 26 cm. How wide is it?

  • The length is \(w + 3\). Going round: \(w + (w + 3) + w + (w + 3) = 4w + 6\).
  • So \(4w + 6 = 26\). Take 6 from both sides: \(4w = 20\). Divide by 4: \(w = 5\).
  • The rectangle is 5 cm wide and 8 cm long. Check: \(5 + 8 + 5 + 8 = 26\).

Try it

  1. Find the value of \(5n - 3\) when \(n = 6\).
  2. Simplify \(7y + 4 - 2y + 9\).
  3. Solve \(x - 8 = 13\).
  4. Solve \(4m + 5 = 33\).
  5. A cinema ticket costs \(t\) dollars and popcorn costs 6 dollars. Three friends each buy a ticket and they share one popcorn. Write an expression for the total cost. If they pay 42 dollars in all, what does one ticket cost?
  6. Explain, with algebra, this trick: "Think of a number, add 3, multiply by 2, take away 6, halve it. You are back where you started."
Answers
  1. \(5 \times 6 - 3 = 30 - 3 = 27\).
  2. \(7y - 2y = 5y\) and \(4 + 9 = 13\), so the answer is \(5y + 13\).
  3. Add 8 to both sides: \(x = 21\). Check: \(21 - 8 = 13\).
  4. Take 5 from both sides: \(4m = 28\). Divide by 4: \(m = 7\). Check: \(28 + 5 = 33\).
  5. Total \(= 3t + 6\). Then \(3t + 6 = 42\), so \(3t = 36\) and \(t = 12\). One ticket costs 12 dollars.
  6. Start with \(n\). Add 3: \(n + 3\). Multiply by 2: \(2n + 6\). Take away 6: \(2n\). Halve it: \(n\). You end with the number you started with, whatever it was.

Common mistake

Doing something to only one side. In \(x + 3 = 7\), crossing out the 3 on the left without taking 3 from the right gives \(x = 7\), which is wrong (check: \(7 + 3 = 10\), not 7). Every step must be done to both sides, or the scale tips. A second classic slip: thinking \(3n + 5\) equals \(8n\). It does not; \(3n\) and \(5\) are not like terms. Test it with \(n = 2\): \(3 \times 2 + 5 = 11\), but \(8 \times 2 = 16\).

Where it leads