Arithmetic with Expressions

Intuition

You spend most of arithmetic collapsing a messy expression into a single tidy number: $3 + 4 \times 2$ becomes $11$ , and there is nothing left to say. Once a variable is present you usually cannot reach a single number, because the value depends on the variable. What you can still do, and what this lesson is about, is tidy the expression up, rewriting $3x + 2 + 5x$ as the shorter $8x + 2$ . The shorter form names the same value for every $x$ , but it is far easier to read and use. That tidying has two moves: gathering terms of the same kind, and handling the plus and minus signs between whole expressions correctly. Both are just the laws from the first lesson doing quiet work, so every rule you need is one you have already met. Here you simply push it harder.

Learning goals

Name a term's coefficient and variable part
Combine like terms by adding coefficients, the distributive law reversed
Add expressions by dropping parentheses and collecting
Subtract by adding the opposite, flipping every sign inside
Scale an expression by distributing across every term
Check a simplification by substituting a value into both forms

The parts of an expression

An expression is any meaningful combination of numbers, variables, and operations that names a value, such as $3x^2 - 5x + 7$ . Its building blocks have names worth fixing now, because the rest of the lesson is about moving these blocks around without breaking anything.

A term is a piece of the expression that is added to the others. The expression $3x^2 - 5x + 7$ has three terms: $3x^2$ , $-5x$ , and $7$ . Because subtraction is adding the opposite, the sign in front of a term belongs to it, so the middle term is $-5x$ , not $5x$ . Keeping the sign attached to its term is what makes the arithmetic below reliable.

The coefficient of a term is its numerical factor, and the variable part is what is left. In $-5x$ the coefficient is $-5$ and the variable part is $x$ ; in $3x^2$ the coefficient is $3$ and the variable part is $x^2$ . A term like $x$ or $x^2$ that shows no number has an invisible coefficient of $1$ , since $x = 1x$ . A constant term is a term with no variable at all, just a number, like the $7$ above.

Finally, like terms are terms with the same variable part, meaning the same variables raised to the same powers. So $3x$ and $-5x$ are like terms, and so are $4xy$ and $7yx$ , because $yx = xy$ makes their variable parts identical. But $3x^2$ and $5x$ are unlike, because $x^2$ and $x$ are different variable parts, and $3x$ and $3y$ are unlike because they use different letters. Telling like from unlike is the whole game, so read the variable part carefully every time.

Combining like terms

To simplify an expression is to rewrite it as an equivalent expression with as few terms as possible, and the main tool is combining like terms. The rule is short: like terms combine into one term by adding their coefficients, and unlike terms cannot combine at all. Both halves of that rule come straight from the distributive law you proved in the first lesson.

Why like terms combine, and unlike terms do not#

The distributive law says $a(b + c) = ab + ac$ . Read it from right to left and it becomes a rule for pulling out a shared factor: $ab + ac = a(b + c)$ . That reverse reading is the entire engine behind combining like terms.

Two like terms share the same variable part. Write that shared part as $v$ , so the two terms are $mv$ and $nv$ for some numbers $m$ and $n$ . For $3x$ and $5x$ the shared part is $v = x$ with $m = 3$ and $n = 5$ ; for $4x^2$ and $x^2$ it is $v = x^2$ . Because multiplication is commutative, $mv = vm$ and $nv = vn$ , so

mv + nv = vm + vn = v(m + n) = (m + n)v.

The two terms collapse into a single term whose coefficient is the sum $m + n$ and whose variable part is unchanged. This is why $3x + 5x = (3 + 5)x = 8x$ , and why $x + x = 1x + 1x = (1 + 1)x = 2x$ , since a bare $x$ carries the invisible coefficient $1$ .

Unlike terms have different variable parts, so there is no single $v$ to factor out and the step above never starts. In $3x + 5$ one term has variable part $x$ and the other is a constant; in $3x + 3y$ the parts are $x$ and $y$ . In $3x + 3x^2$ the variable parts are $x$ and $x^2$ . None of these pairs shares a variable part, so each sum is already as simple as it gets and must be left as two separate terms.

It helps to picture terms as tiles. Let a long tile stand for one $x$ and a small square for a $1$ . Then combining like terms is nothing more than sorting the tiles into piles of the same shape and counting each pile.

Picture each long tile as one x and each small tile as 1. The top row shows 3x + 2 + 2x + 3 in the order written; sorting the tiles into a pile of x-tiles and a pile of unit-tiles and counting each pile gives 5x + 5. You can gather the x-tiles together, but you cannot merge an x-tile with a unit-tile, which is exactly why unlike terms never combine.

Worked example 1 Name the parts, then combine $4x^2 - x + 7 - 2x^2 + 3x$

First read off the terms with their signs: $4x^2$ , $-x$ , $7$ , $-2x^2$ , and $3x$ . The constant term is $7$ . The like terms fall into two families, the $x^2$ terms $4x^2$ and $-2x^2$ , and the $x$ terms $-x$ and $3x$ .

Combine each family by adding coefficients. For the $x^2$ terms,

4x^2 - 2x^2 = (4 - 2)x^2 = 2x^2,

and for the $x$ terms, remembering that $-x$ has coefficient $-1$ ,

-x + 3x = (-1 + 3)x = 2x.

The constant $7$ has nothing to pair with, so the simplified expression is

2x^2 + 2x + 7.

Its three terms are pairwise unlike, so no further combining is possible.

Check your understanding

Combine like terms: $6y + 2 - 4y + 5$ .

2y + 7

2y - 3

10y + 7

2y + 3

Adding and subtracting whole expressions

Often you need to add or subtract entire expressions, each wrapped in parentheses, such as $(4x + 3) + (2x + 5)$ . Adding is the easy case. The associative and commutative laws of addition, which hold for every number, let you erase the parentheses and reorder the terms however you like. So when you are adding, you simply drop the parentheses and combine like terms:

(4x + 3) + (2x + 5) = 4x + 3 + 2x + 5 = (4x + 2x) + (3 + 5) = 6x + 8.

Subtraction takes one extra moment of care, because subtracting an expression is not the same as subtracting only its first term. Subtracting a whole expression means adding its opposite, and the opposite of a sum flips the sign of every term inside.

Why a minus sign in front reverses every term#

To subtract an expression is to add its opposite, and the opposite of any quantity $Q$ is $(-1) \cdot Q$ . So a minus sign written in front of a parenthesis is really the factor $-1$ waiting to be distributed.

Take $-(3x - 5)$ . Rewrite the subtraction inside as an addition, $3x - 5 = 3x + (-5)$ , then distribute the $-1$ across both terms:

-(3x - 5) = (-1)\big(3x + (-5)\big) = (-1)(3x) + (-1)(-5) = -3x + 5.

Every term inside came out with its sign reversed: $3x$ became $-3x$ , and $-5$ became $+5$ . Nothing here depends on the particular numbers, so the same reasoning gives the general rule $-(a + b) = -a - b$ . Distributing a leading minus sign changes the sign of each term it reaches. Forgetting to flip the second term is the single most common error in this whole lesson.

Worked example 2 Subtract $(3x - 5)$ from $(7x + 2)$

Subtraction adds the opposite of the second expression, so distribute the minus sign across both of its terms:

(7x + 2) - (3x - 5) = 7x + 2 - 3x + 5.

The $-5$ inside became $+5$ ; missing that flip is the classic slip. Now reorder and combine like terms:

7x - 3x + 2 + 5 = 4x + 7.

To check, substitute a value into both the original and the result and confirm they agree. At $x = 3$ the original is $(7(3) + 2) - (3(3) - 5) = 23 - 4 = 19$ , and the simplified form is $4(3) + 7 = 19$ . They match, which is the reassurance that the two expressions are equivalent.

Check your understanding

Simplify $(5x - 3) - (2x - 8)$ .

3x - 11

3x + 5

7x + 5

3x - 5

Scaling an expression by a number or a term

You already know the distributive law, $a(b + c) = ab + ac$ . It is what lets you scale an entire expression by a single factor: multiply each term inside the parentheses by the factor outside. Scaling $3x + 1$ by $2$ gives

2(3x + 1) = 2(3x) + 2(1) = 6x + 2,

and the factor can carry a sign, so scaling by $-2$ would give $-6x - 2$ . The factor outside can even be a single term rather than a bare number; distributing $x$ across $2x + 3$ gives $x(2x + 3) = 2x^2 + 3x$ . Multiplying two multi-term expressions together is a bigger job, saved for the next lesson. Most simplifications here combine scaling with the earlier skills. Distribute every product first, then collect like terms, and when grouping symbols are nested inside one another, clear those symbols from the inside out.

Worked example 3 Distribute, then combine $2(3x + 1) + 3(x - 4)$

Use the distributive law on each product first, keeping every sign:

2(3x + 1) = 6x + 2, \qquad 3(x - 4) = 3x - 12.

Now the expression is a plain sum, so gather like terms:

6x + 2 + 3x - 12 = (6x + 3x) + (2 - 12) = 9x - 10.

The two steps never blur together: distribute every product before you start collecting like terms.

Worked example 4 Simplify the nested expression $8x - 2[3x - (x - 4)]$

Nested grouping symbols are cleared from the inside out. Start with the innermost parentheses, where a minus sign flips both terms:

3x - (x - 4) = 3x - x + 4 = 2x + 4.

Multiply by the $2$ outside the brackets:

2[2x + 4] = 4x + 8.

Finally subtract that whole quantity from $8x$ , flipping its signs:

8x - (4x + 8) = 8x - 4x - 8 = 4x - 8.

Reaching outward before an inner layer is finished is what produces sign mistakes, so resolve one bracket completely before moving out to the next.

Check your understanding

Simplify $3(2x - 1) - (x - 4)$ .

5x - 1

7x + 1

5x + 1

5x - 7

Checking a simplification by substitution

Simplifying is supposed to produce an equivalent expression, one that gives the same value for every input. So a quick substitution is the natural way to catch an error. Pick a convenient value, evaluate both the original and your simplified form, and compare. Agreement does not prove the two are equivalent (recall from the first lesson that no finite list of checks can). But a disagreement instantly reveals a slip, almost always a dropped sign. Values like $x = 2$ or $x = -1$ tend to expose mistakes that $x = 0$ or $x = 1$ quietly hide.

Worked example 5 Simplify and check $2(x^2 + 3x) - (x^2 - x)$

Distribute the $2$ , then subtract every term of the second group:

2(x^2 + 3x) - (x^2 - x) = 2x^2 + 6x - x^2 + x.

Combine the $x^2$ terms and the $x$ terms:

(2x^2 - x^2) + (6x + x) = x^2 + 7x.

Now verify with a substitution, which a correct simplification will survive. At $x = 2$ the original is $2(4 + 6) - (4 - 2) = 20 - 2 = 18$ , and $x^2 + 7x$ gives $4 + 14 = 18$ . The agreement is the sign that every step preserved the value, which is exactly what simplifying is meant to do.

Common mistakes

Adding unlike terms. $3x + 5$ is already fully simplified; it is not $8x$ . Only terms with the same variable part combine, so a variable term and a constant stay apart.
Forgetting to flip every sign after a minus. $(7x + 2) - (3x - 5) = 4x + 7$ , not $4x - 3$ . The $-5$ becomes $+5$ , because subtracting an expression negates all of its terms.
Distributing to only the first term. $2(3x + 1) = 6x + 2$ , not $6x + 1$ . The factor must reach every term inside the parentheses.
Losing the coefficient of $1$ . A lone variable counts as one copy, so $x + x = 2x$ and $7a - a = 6a$ , never $x$ or $7a$ .
Merging different powers. $x^2$ and $x$ are unlike terms, so $2x^2 + 3x$ does not become $5x^2$ or $5x^3$ ; leave the two powers separate.
Changing the value while “simplifying.” Combining like terms must leave an equivalent expression. If a substitution check disagrees, a term was dropped or a sign was flipped somewhere.

Key takeaways

A term is a signed piece of an expression; its coefficient is the number factor and its variable part is the rest. A constant term has no variable, and a lone variable has coefficient $1$ .
Like terms share the same variable part (same letters, same powers). They combine by adding coefficients, $3x + 5x = (3 + 5)x = 8x$ , which is the distributive law read in reverse. Unlike terms such as $3x + 5$ , $3x + 3y$ , and $3x + 3x^2$ do not combine.
To add expressions, drop the parentheses and collect like terms; the commutative and associative laws make the reordering legal.
To subtract an expression, add its opposite, so a leading minus sign flips the sign of every term inside: $(7x + 2) - (3x - 5) = 4x + 7$ .
To scale by a number or a single term, distribute it across every term: $2(3x + 1) = 6x + 2$ . Clear nested grouping symbols from the inside out.
A substitution check guards your work: an equivalent expression agrees with the original at every value, so any disagreement flags a mistake.

Arithmetic with Expressions

Learning goals

The parts of an expression

Combining like terms

Adding and subtracting whole expressions

Scaling an expression by a number or a term

Checking a simplification by substitution

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Free Response Questions (FRQ)