Arithmetic with Expressions

Learning goals

Name a term's coefficient and variable part, and tell like terms from unlike
Combine like terms by adding coefficients, the distributive law reversed
Add and subtract expressions by dropping parentheses and flipping signs
Scale an expression by distributing across every term, including nested grouping
Check a simplification by substitution, and say what agreement and disagreement each show

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. Two constant terms, such as $7$ and $-2$ , are like terms too, since neither one has a variable part to differ on. Telling like from unlike is the whole game, so read the variable part carefully every time.

Check your understanding

What are the coefficient and variable part of the term $-6x^3$ ?

A Coefficient

-6

, variable part

x^3

B Coefficient

6

, variable part

x^3

C Coefficient

-6

, variable part

x

D Coefficient

-6x

, variable part

3

Combining like terms

To simplify an expression is to rewrite it as an equivalent expression that has no more like terms left to combine, and the main tool is combining like terms. The rule is short: like terms combine into one term by adding their coefficients; terms with different variable parts cannot combine that way.

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 why unlike terms never combine this way.

The sorting in the picture is exactly what the distributive law says in reverse. Here is that same idea written with algebra, so you can see why the tile trick always works.

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 do not share a complete variable part, so there is no single $v$ whose coefficient the two terms could combine around, 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 $x^2$ , even though both terms do carry a factor of $x$ . None of these pairs shares a complete variable part, so none of them can be merged into one term by adding coefficients. A pair like $3x + 3x^2$ can still be factored a different way later on, but that is a different move from combining like terms.

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 $3x$ .

Term	Coefficient	Variable part
$4x^2$	$4$	$x^2$
$-x$	$-1$	$x$
$7$	$7$	none (constant)
$-2x^2$	$-2$	$x^2$
$3x$	$3$	$x$

Two like-term families show up: the $x^2$ terms $4x^2$ and $-2x^2$ , and the $x$ terms $-x$ and $3x$ . The $7$ is the constant term, with nothing to pair with.

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 a good sign the simplification is correct.

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 lets you scale an entire expression by a single factor: multiply every term inside the parentheses by the factor outside. Scaling $3x + 1$ by $2$ gives

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

The factor can carry a sign, and the sign travels to every term it reaches. Scaling $3x + 1$ by $-2$ instead gives

-2(3x + 1) = (-2)(3x) + (-2)(1) = -6x - 2.

The factor outside can even be a single term rather than a bare number. Distributing $x$ across $2x + 3$ still reaches every term, multiplying powers as it goes:

x(2x + 3) = x(2x) + x(3) = 2x^2 + 3x,

since $x \cdot 2x = 2(x \cdot x) = 2x^2$ . Multiplying two multi-term expressions together, where both factors have more than one term, is a bigger job saved for the next lesson.

When a simplification nests grouping symbols inside one another, distribute one layer at a time and clear them from the inside out: finish everything inside the innermost parentheses before touching the symbol around it. Most simplifications from here combine scaling with the earlier skills, so distribute every product first and only then collect like terms.

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 the nested expression $5x - 2[3x - (x - 1)]$ .

x - 2

x + 2

x - 1

9x + 2

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 at one chosen value is evidence the two are equivalent, not proof of it. But a disagreement at any value 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 good evidence that the simplified form is equivalent; a mismatch, not a match, is what would have settled the question for certain.

Check your understanding

A classmate simplifies $3(x + 4) - (x - 1)$ and gets $2x + 13$ . At $x = 2$ , both the original and $2x + 13$ equal $17$ . What does that agreement show?

A It proves the simplification is correct for every value of

x

B It is good evidence the simplification is correct, but it does not prove that for every

x

C It proves nothing at all, since only one value was tried D It proves the simplification is wrong

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, some step changed the expression, most often a dropped term or a flipped sign.

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 $3x + 3x^2$ cannot be merged into one term this way.
To add expressions, drop the parentheses and collect like terms. To subtract an expression, add its opposite, so a leading minus sign flips the sign of every term inside: $(7x + 2) - (3x - 5) = 4x + 7$ . The commutative and associative laws make the reordering legal.
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. A single agreement is good evidence, not proof, since it tests only one value out of infinitely many.

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)

Core practice