Writing Algebraic Expressions

An algebraic expression uses letters to stand for numbers, like $3n + 1$. Expressions let you describe a whole pattern in one short line, instead of listing case after case. On this page you’ll learn the vocabulary, turn words and patterns into expressions, and check whether two different-looking expressions really say the same thing.

Key ideas

Variables, terms, coefficients and constants

A variable is a letter that stands for a number that can change, like $n$ or $x$.

Take the expression $5x - 2y + 7$. It has three terms, separated by $+$ and $-$ signs:

Part	In $5x - 2y + 7$	What it means
Terms	$5x$, $-2y$, $7$	the pieces being added
Coefficient of $x$	$5$	the number multiplying $x$
Coefficient of $y$	$-2$	the sign belongs to the term
Constant	$7$	a term with no variable

Two small but important facts:

• $5x$ means $5 \times x$. We don’t write the multiplication sign.
• $x$ on its own has a coefficient of $1$, and $-x$ has a coefficient of $-1$.

Translating words into expressions

Let $n$ be the number. Here are the most common phrases:

Words	Expression
5 more than a number	$n + 5$
3 less than a number	$n - 3$
twice a number	$2n$
a number divided by 4	$\dfrac{n}{4}$
3 less than twice a number	$2n - 3$
twice the sum of a number and 3	$2(n + 3)$

Watch out for “less than”. “3 less than a number” starts with the number and takes 3 away, so it’s $n - 3$, not $3 - n$.

Also watch for “the sum” or “the difference”. They tell you to add or subtract first, so you need brackets: “twice the sum of a number and 3” is $2(n + 3)$.

Generalizing a growing pattern

Here is a pattern made of toothpicks. Each figure adds one more square.

Each new square needs 3 more toothpicks, so Figure $n$ uses $3n + 1$ toothpicks.

Put the counts in a table:

Figure number, $n$	1	2	3	4
Toothpicks	4	7	10	13

The count goes up by 3 each time, so the expression starts with $3n$. But $3n$ gives $3$ for Figure 1, and we need $4$. Add $1$ to fix it: $3n + 1$.

The picture tells you why. Start with the orange toothpick on the left. Each square then adds three blue ones (top, bottom and right side). So $n$ squares need $3n$ blue toothpicks plus $1$ orange one: $3n + 1$.

Now you can find any figure without drawing it. Figure 50 uses $3(50) + 1 = 151$ toothpicks.

Equivalent expressions

Two expressions are equivalent if they give the same value for every value of the variable. For example, $2(x + 3)$ and $2x + 6$ are equivalent. There are three ways to check:

1. Substitute numbers. Try several values of $x$ in both expressions and compare.
2. Graph them. Graph $y =$ each expression. Equivalent expressions give exactly the same line or curve.
3. Use algebra. Simplify one expression until it looks like the other. Here, $2(x + 3) = 2x + 6$ by the distributive property. You’ll practise this in simplifying algebraic expressions.

$y = 2(x + 3)$ and $y = 2x + 6$ are the same line, so the expressions are equivalent. $y = 2x + 3$ is a different line.

Why one number isn’t enough. Try $x = 2$ in $2x$ and $x^2$. Both give $4$! But try $x = 3$: $2x = 6$ and $x^2 = 9$. They’re not equivalent. A matching value can be a coincidence, so substituting can only suggest that two expressions are equivalent. A single value that doesn’t match does prove they are not equivalent. To be sure they are equivalent, use algebra.

Where this comes from

The word “algebra” comes from the Arabic al-jabr, part of the title of a book written in Baghdad around 820 CE by the mathematician Muhammad ibn Musa al-Khwarizmi. He solved problems entirely in words, with no letters for unknowns. Using letters like $x$ became common much later, in Europe in the 1600s. Today, the same idea runs every spreadsheet formula and computer program you use.

Worked examples

Example 1: Naming the parts

For the expression $7x^2 - x + 4y - 10$, list the terms, the coefficient of $x$, the coefficient of $y$, and the constant.

Solution.

• Terms: $7x^2$, $-x$, $4y$ and $-10$. There are four terms.
• Coefficient of $x$: the term is $-x$, which means $-1x$, so the coefficient is $-1$.
• Coefficient of $y$: $4$.
• Constant: $-10$ (keep the sign with the term).

Example 2: From words to an expression

Write an expression for each. Let $n$ be the number.

• (a) 6 less than three times a number
• (b) the product of 5 and the sum of a number and 2
• (c) the cost, in dollars, of $n$ movie tickets at $14 each plus one $9 popcorn

Solution.

(a) “Three times a number” is $3n$. “6 less than” that means take 6 away: $3n - 6$.

(b) “The sum of a number and 2” is $n + 2$. The product of 5 and that whole sum needs brackets: $5(n + 2)$.

(c) $n$ tickets at 14 dollars each cost $14n$ dollars. Add the popcorn: $14n + 9$.

Check (c): 2 tickets and a popcorn cost $14(2) + 9 = 37$ dollars. That’s $28 + 9$, which makes sense. ✓

Example 3: A tile pattern from a table

A tile pattern grows like this:

Figure number, $n$	1	2	3	4
Tiles	5	9	13	17

• (a) Write an expression for the number of tiles in Figure $n$.
• (b) How many tiles are in Figure 20?
• (c) Which figure uses 101 tiles?

Solution.

(a) The number of tiles goes up by 4 each time, so start with $4n$. For Figure 1, $4n = 4$, but we need 5. Add 1: the expression is $4n + 1$.

Check Figure 3: $4(3) + 1 = 13$. ✓

(b) $4(20) + 1 = 80 + 1 = 81$ tiles.

(c) We need $4n + 1 = 101$. Take away 1: $4n = 100$. Divide by 4: $n = 25$. It’s Figure 25.

Example 4: Are they equivalent?

Which of these are equivalent to $3(x - 2) + x$: $\;4x - 6\;$ or $\;4x - 2$?

Solution.

Substitute. Try a few values:

$x$	$3(x - 2) + x$	$4x - 6$	$4x - 2$
0	$3(-2) + 0 = -6$	$-6$	$-2$
1	$3(-1) + 1 = -2$	$-2$	$2$
5	$3(3) + 5 = 14$	$14$	$18$

$4x - 2$ already fails at $x = 0$, so it is not equivalent. $4x - 6$ matches every time, which suggests it is equivalent, but three checks aren’t proof.

Use algebra. Multiply out the bracket, then combine the $x$ terms:

$$3(x - 2) + x = 3x - 6 + x = 4x - 6$$

So $3(x - 2) + x$ and $4x - 6$ are equivalent. (If you graphed them, you would get the same line.)

Common mistakes

Writing “less than” backwards. “4 less than a number” is $n - 4$, not $4 - n$. Test it with a real number: 4 less than 10 is 6, and $10 - 4 = 6$.

Forgetting brackets. “Three times the sum of a number and 5” is $3(n + 5)$. Writing $3n + 5$ only multiplies the number by 3, not the 5. When you see “the sum” or “the difference”, you almost always need brackets.

Dropping the sign of a term. In $8 - 3x$, the coefficient of $x$ is $-3$, not $3$. The sign in front of a term belongs to that term.

Using only the “jump” in a pattern. If a pattern goes 4, 7, 10, 13, the expression is not just $3n$. Always check your expression with Figure 1: $3(1) = 3$, but there are 4 toothpicks, so you need $3n + 1$.

Trusting one matching value. $2x$ and $x^2$ both equal 4 when $x = 2$, but they aren’t equivalent. Test several values, including $0$ and a negative number, and use algebra to be sure.

Practice

1. (Warm-up) For the expression $-2p + 9q - 1$, list the terms, the coefficient of $p$, the coefficient of $q$, and the constant.

Solution

Terms: $-2p$, $9q$ and $-1$.

Coefficient of $p$: $-2$. Coefficient of $q$: $9$. Constant: $-1$.

2. (Warm-up) Write an expression for each. Let $n$ be the number.

• (a) 10 more than a number
• (b) 7 less than a number
• (c) a number divided by 3, minus 4
• (d) twice the difference of a number and 5

Solution

(a) $n + 10$

(b) $n - 7$

(c) $\dfrac{n}{3} - 4$

(d) “The difference of a number and 5” is $n - 5$. Twice that is $2(n - 5)$.

3. (Warm-up) In the toothpick pattern above, Figure $n$ uses $3n + 1$ toothpicks. How many toothpicks are in Figure 10? Describe Figure 10 in words.

Solution

$$3(10) + 1 = 30 + 1 = 31$$

Figure 10 is a row of 10 squares, made of 31 toothpicks.

4. (Core) A phone plan costs $25 per month, plus $10 for each extra gigabyte (GB) of data.

• (a) Write an expression for the monthly cost, in dollars, if you use $d$ extra GB.
• (b) What does the plan cost in a month when you use 3 extra GB?

Solution

(a) Each extra GB adds 10 dollars, so $d$ extra GB add $10d$. The monthly cost is $25 + 10d$ (or $10d + 25$).

(b) $25 + 10(3) = 25 + 30 = 55$. The plan costs $55 that month.

5. (Core) Here is a pattern of dots:

Figure number, $n$	1	2	3	4
Dots	2	7	12	17

• (a) Write an expression for the number of dots in Figure $n$.
• (b) How many dots are in Figure 50?

Solution

(a) The count goes up by 5 each time, so start with $5n$. For Figure 1, $5n = 5$, but there are only 2 dots, so subtract 3. The expression is $5n - 3$.

Check Figure 4: $5(4) - 3 = 17$. ✓

(b) $5(50) - 3 = 250 - 3 = 247$ dots.

6. (Core) A row of triangles is made from toothpicks. One triangle uses 3 toothpicks, two triangles (sharing a side) use 5, and three triangles use 7.

• (a) Write an expression for the number of toothpicks in a row of $n$ triangles. Explain it using the picture.
• (b) How many triangles can you make in a row with 61 toothpicks?

Solution

(a) The count goes up by 2 each time: each new triangle shares one side with the one before it, so it needs only 2 new toothpicks. Start with 1 toothpick, and add 2 for each triangle: $2n + 1$.

Check: $n = 3$ gives $2(3) + 1 = 7$. ✓

(b) Solve $2n + 1 = 61$. Take away 1: $2n = 60$. Divide by 2: $n = 30$. You can make 30 triangles.

7. (Core) Decide whether each pair of expressions is equivalent. Justify your answer.

• (a) $2(x + 4)$ and $2x + 4$
• (b) $5x - x$ and $4x$
• (c) $x \cdot x$ and $2x$

Solution

(a) Not equivalent. Try $x = 0$: $2(0 + 4) = 8$, but $2(0) + 4 = 4$. One value that doesn’t match is enough to prove they’re different. (The 2 must multiply both $x$ and $4$: $2(x + 4) = 2x + 8$.)

(b) Equivalent. Five $x$‘s take away one $x$ leaves four $x$‘s: $5x - x = 4x$ for every value of $x$.

(c) Not equivalent. Try $x = 3$: $x \cdot x = 9$, but $2x = 6$. (Be careful: they do match at $x = 0$ and $x = 2$, which is why you should test more than one value.)

8. (Challenge) Priya tests the expressions $x^2 + 2$ and $3x$. When $x = 1$, both give $3$. When $x = 2$, both give $6$. She says they must be equivalent. Is she right? Explain.

Solution

No. Try $x = 3$:

$$x^2 + 2 = 9 + 2 = 11, \qquad 3x = 3(3) = 9$$

They give different values, so they are not equivalent. Matching at two values was a coincidence. On a graph, $y = x^2 + 2$ is a curve and $y = 3x$ is a line; they just happen to cross at $x = 1$ and $x = 2$. Equivalent expressions must match for every value, and their graphs would be the same.

9. (Challenge) A square frame is made of square tiles around the edge of an $n$ by $n$ square (with $n \ge 2$). For example, a 5 by 5 frame has a border of 16 tiles. Two students count the tiles differently:

• Amir writes $4(n - 1)$.
• Bea writes $2n + 2(n - 2)$.

Explain how each student might have counted, and show that their expressions are equivalent.

Solution

Amir: go around the frame in four equal pieces. Each piece is one side of $n$ tiles, minus the corner where the next side starts, so each piece has $n - 1$ tiles. Four pieces: $4(n - 1)$.

Bea: count the whole top row and bottom row ($n$ tiles each, so $2n$). The left and right columns are left, and each is missing its top and bottom tile, so each has $n - 2$ tiles: $2(n - 2)$. Total: $2n + 2(n - 2)$.

Equivalent? Use the distributive property on both:

$$4(n - 1) = 4n - 4$$$$2n + 2(n - 2) = 2n + 2n - 4 = 4n - 4$$

Both simplify to $4n - 4$, so they’re equivalent. Check with $n = 5$: $4(4) = 16$ and $10 + 2(3) = 16$. ✓