Evaluating and Simplifying Expressions

Intuition

There are two very different things you can do with an expression, and this lesson is about both. You can evaluate it: hand the variables actual numbers and work out the single value that comes out, turning $4x + 7$ at $x = 5$ into $27$ . Or you can simplify it: rewrite it as a shorter expression that always means the same thing, for every possible input, without ever choosing a number for $x$ . That move turns $3x + 5x$ into $8x$ . Evaluating always ends in a single number. Simplifying usually still carries a variable, but not always: it is possible to simplify all the way down to a plain number, as $x - x$ does to $0$ . Evaluating leans on the order of operations; simplifying leans on the distributive property and combining like terms. Both are tools you already own.

Learning goals

Evaluate with the full order of operations, protecting each substituted sign with parentheses
Combine like terms by adding coefficients, and explain why using the distributive property run backward
Distribute first when parentheses are present, then collect, treating a leading minus as $-1$
Distinguish simplifying from evaluating, and say what each produces

Evaluating with the full order of operations

You met substitution in the last lesson: replace each variable with its value, written in parentheses, then compute. With a single small term that is the whole story. The moment an expression has several terms, an exponent, or a grouped piece, the second half of the job starts doing real work. That second half is the order of operations.

Recall the order from the order-of-operations chapter, often remembered as PEMDAS: first whatever is inside Parentheses, then Exponents. Next come Multiplication and Division from left to right, and last Addition and Subtraction from left to right. Substitution does not change any of that. It only swaps a letter for a number; the operations already written around that letter still run in their fixed order.

So evaluating a multi-term expression happens in two steps. Step one: substitute every variable, each value wrapped in its own parentheses. Step two: work through the resulting arithmetic strictly in order, one operation at a time. Wrapping each value in parentheses in step one protects its sign through step two.

Worked example 1 Evaluate $3x^2 - 5x + 8$ when $x = 4$

Substitute $4$ for every $x$ , each one in its own parentheses, and copy the rest of the expression unchanged:

3x^2 - 5x + 8 = 3(4)^2 - 5(4) + 8.

Now work in order. Exponents come before multiplication, so square the $4$ first:

3(4)^2 = 3 \cdot 16 = 48.

The middle term is a multiplication, $5(4) = 20$ . Substituting those two results back leaves only additions and subtractions:

48 - 20 + 8.

Going left to right, $48 - 20 = 28$ , then $28 + 8 = 36$ . So the expression is worth $36$ when $x = 4$ . The exponent reached only the $4$ , never the coefficient $3$ , because $x^2$ squares the variable alone.

When the input is negative

A negative input is where the parentheses earn their keep. The whole point of writing a substituted value as $(-2)$ instead of a bare $-2$ is to keep the minus sign glued to the number. That glue matters when an exponent or a coefficient acts on the number. Two cases trip people up constantly.

The first is squaring a negative. If $x = -3$ , then $x^2$ means $(-3)^2$ , and a negative times a negative is positive, so $(-3)^2 = 9$ . Without the parentheses you might write $-3^2$ , which by the order of operations means $-(3^2) = -9$ , the wrong sign entirely. The parentheses are the difference between $9$ and $-9$ .

The second is a coefficient meeting a negative. In $-5x$ with $x = -3$ you get $-5(-3) = 15$ , because the two minus signs multiply to a plus. Keep both signs visible and let the integer rules from chapter 2 settle them.

Worked example 2 Evaluate $2x^2 - 4x - 1$ when $x = -3$

Substitute $-3$ for each $x$ , every value inside parentheses so the sign cannot drift:

2x^2 - 4x - 1 = 2(-3)^2 - 4(-3) - 1.

Exponents first. Squaring $-3$ gives a positive result, since a negative times a negative is positive:

(-3)^2 = 9, \qquad \text{so} \qquad 2(-3)^2 = 2 \cdot 9 = 18.

Next the multiplication in the middle term. Watch the sign already in front: the term is $-4x$ , so

-4(-3) = +12.

Substituting both results back, the expression becomes a short arithmetic chain:

18 + 12 - 1 = 29.

So the value is $29$ . Every sign survived because each $-3$ traveled inside its own parentheses.

Check your understanding

Evaluate $x^2 + 3x$ when $x = -4$ .

4

28

-4

-28

Two variables and grouped pieces

Nothing changes when there are two letters: substitute both, still each in parentheses, then compute in order. A grouped piece inside parentheses must be finished first, exactly as the “P” in PEMDAS demands, before it is combined with anything else in the expression.

Worked example 3 Evaluate $5(a + b) - a^2$ when $a = 2$ and $b = 6$

Replace $a$ with $2$ and $b$ with $6$ , keeping the grouping parentheses that were already there:

5(a + b) - a^2 = 5\big((2) + (6)\big) - (2)^2.

The order of operations says finish the inside of the parentheses first:

(2) + (6) = 8, \qquad \text{so the first term is} \qquad 5(8) = 40.

Then the exponent in the last term, $(2)^2 = 4$ . The line is now

40 - 4 = 36.

So the expression equals $36$ . The addition $a + b$ ran before the multiplication by $5$ only because parentheses surrounded it. Without them, the order of operations would multiply first and change the answer.

Simplifying: the idea of like terms

Now set substitution aside. To simplify an expression is to rewrite it in a shorter, equivalent form, one that produces the same value for every possible input. You do that rewriting without choosing any input at all. The main tool for doing this is combining like terms.

Two terms are like terms when they have exactly the same variable part: the same letters raised to the same powers. Only the coefficient out front is allowed to differ. The terms $3x$ and $5x$ are like terms, both being some number of $x$ ‘s. The terms $7y$ and $2y$ are like terms. But $4x$ and $4y$ are not like terms, because the variables differ, and $x$ and $x^2$ are not like terms, because the powers differ. A plain number, a constant, is like only another constant.

Three $x$ ‘s plus five $x$ ‘s is eight $x$ ‘s, the same way three apples plus five apples is eight apples: you are just counting how many copies of the same thing you have. The reason that always works, no matter what the coefficients are, comes straight from the distributive property you proved in chapter 1.

Why $3x + 5x = 8x$ #

Recall the distributive property, $a(b + c) = ab + ac$ . Read from right to left it says that a common factor can be pulled out of a sum: $ab + ac = a(b + c)$ . The two terms $3x$ and $5x$ share exactly such a common factor, namely $x$ , since $3x = x \cdot 3$ and $5x = x \cdot 5$ .

Pull that shared $x$ out front. The distributive property, run backward, gives

3x + 5x = x(3 + 5).

Inside the parentheses is ordinary arithmetic: $3 + 5 = 8$ . Therefore

3x + 5x = x(8) = 8x.

So combining like terms is not a new rule to memorize. It is the distributive property pulling the whole matching variable part out front, then adding the coefficients left behind. That only works when the two variable parts are identical, letter for letter and power for power. $3x$ and $5y$ do not match, so there is no single variable part to pull out, and the distributive property gives nothing useful. The sum $3x + 5y$ is already as short as it gets.

Sharing part of a variable part is not enough either. $3x$ and $5x^2$ both contain a factor of $x$ , but $x$ and $x^2$ are different variable parts, so they stay unlike terms and cannot combine, even though $x$ divides evenly into both.

This gives you a shortcut you will use every day. To combine like terms, add or subtract their coefficients and keep the common variable part unchanged. The variable part is a label that rides along: you never add the exponents or change the letter.

Check your understanding

Which pair are NOT like terms, even though both terms contain a factor of $x$ ?

3x

and

5x^2

3x

and

5x

7y

and

2y

-4

and

9

Check your understanding

Which equation is the reason $3x + 5x$ is allowed to combine into $8x$ ?

3x + 5x = x(3 + 5)

, the distributive property run backward B

3x + 5x = (3 \cdot 5)x

, since multiplying the coefficients is always allowed C

3x + 5x = 15x^2

, since the exponents add when terms combine D

3x + 5x = 8x^2

, since

x

is written once for both terms

Like terms share the same variable part, so their coefficients add: 3x + 5x collects into 8x. Unlike terms (3x and 5y) have nothing common to pull out, so they stay separate.

Worked example 4 Simplify $7x + 2 + 4x - 9$

Sort the expression into groups of like terms. The variable terms $7x$ and $4x$ are alike; the constants $2$ and $-9$ are alike. Remember each term carries the sign in front of it, so the constants are $+2$ and $-9$ :

7x + 2 + 4x - 9 = (7x + 4x) + (2 - 9).

Each term kept the sign it started with when it moved into its group.

Collect the $x$ -terms by adding their coefficients:

7x + 4x = 11x.

Collect the constants the same way:

2 - 9 = -7.

Put the two simplified groups together:

7x + 2 + 4x - 9 = 11x - 7.

The expression went from four terms to two, and $11x - 7$ has no more like terms to combine, so it is in simplest form. The $x$ -terms and the constants stay separate because they are unlike.

Check your understanding

Simplify $9k - 3 - 4k + 10$ .

5k + 7

13k + 7

5k - 7

5k + 13

Simplifying with the distributive property

Many expressions cannot be simplified until you first remove a set of parentheses, and the tool for that is the distributive property running forward, $a(b + c) = ab + ac$ . Multiply the outside factor by every term inside, then look for like terms to collect. With variables in the mix, the only new feature is that the products keep their letters.

For instance, $3(x + 4)$ distributes to $3 \cdot x + 3 \cdot 4 = 3x + 12$ . The outside $3$ reaches both the $x$ and the $4$ , just as it reached both numbers in the chapter-1 version of the rule. Once the parentheses are gone, any like terms scattered across the expression can be gathered.

Worked example 5 Simplify $2(3x + 5) + 4x$

Start by distributing the $2$ across both terms inside the parentheses:

2(3x + 5) = 2 \cdot 3x + 2 \cdot 5 = 6x + 10.

Rewrite the whole expression with the parentheses gone:

2(3x + 5) + 4x = 6x + 10 + 4x.

Now combine the like terms. The variable terms $6x$ and $4x$ collect, while the constant $10$ has no partner:

6x + 4x = 10x, \qquad \text{so the result is} \qquad 10x + 10.

The simplified expression is $10x + 10$ . Distributing first was essential. Until the parentheses were cleared, there was no $6x$ to pair with the $4x$ .

A minus sign in front of a parenthesis is the classic trap, and it is really just distributing a $-1$ . The expression $-(x - 6)$ means $-1$ times $(x - 6)$ , so the sign of every term inside flips: $-(x - 6) = -x + 6$ . Distributing a negative coefficient works the same way, with the integer sign rules from chapter 2 deciding each product.

Check your understanding

Simplify $-(2x - 5)$ .

-2x + 5

-2x - 5

2x - 5

2x + 5

Worked example 6 Simplify $5(2x - 1) - 3(x - 4)$

Distribute each factor across its parentheses, watching the signs in the second group especially. The first factor, $5$ , gives

5(2x - 1) = 10x - 5.

The second factor is $-3$ , a negative, so it multiplies both inside terms and flips signs accordingly:

-3(x - 4) = (-3)(x) + (-3)(-4) = -3x + 12.

The negative times the negative $4$ became $+12$ . Now write the full expression without parentheses and gather like terms:

5(2x - 1) - 3(x - 4) = 10x - 5 - 3x + 12.

Combine the $x$ -terms and the constants separately:

10x - 3x = 7x, \qquad -5 + 12 = 7,

so the simplified form is $7x + 7$ . The whole result rested on distributing the $-3$ correctly, including the sign change on the $-4$ .

Check your understanding

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

2x + 12

6x + 12

2x + 3

2x + 7

Evaluating versus simplifying

Evaluating and simplifying are distinguished by what you do, not just what you get. Evaluating substitutes a specific value for each variable and always ends in a single number. Simplifying rewrites the expression into an equivalent form that works for every input, without ever substituting a value; that equivalent form usually still carries a variable, but not always, since $x - x$ simplifies to the plain number $0$ even though no value was ever chosen.

They cooperate. Because a simplified expression equals the original for every input, you may simplify once and then evaluate as many times as you need. To evaluate $3(x + 2) + 5x$ at $x = 4$ , simplifying it first to $8x + 6$ turns the work into the single step $8(4) + 6 = 38$ , and that same simplified form makes evaluating at $x = 10$ just as fast: $8(10) + 6 = 86$ .

Two operations on one expression. Simplifying rewrites it as an equivalent expression without substituting a value; evaluating substitutes one and ends in a single number.

Check your understanding

Which word describes turning $6x - 2x$ into $4x$ with no numbers substituted?

A evaluating B simplifying C solving D distributing

Common mistakes

Combining unlike terms. $3x + 4y$ stays as $3x + 4y$ , and $x + x^2$ stays as it is: their variable parts differ, so there is nothing to collect.
Adding the exponents or the variables when combining. $2x + 3x = 5x$ , not $5x^2$ and not $6x$ . Only the coefficients add.
Dropping a sign when substituting a negative. Write $(-3)$ , not a bare $-3$ , so $(-3)^2 = 9$ instead of the careless $-3^2 = -9$ .
Distributing to only the first term. $3(x + 4)$ is $3x + 12$ , never $3x + 4$ . The outside factor reaches every term inside.
Mishandling a minus in front of parentheses. $-(x - 6)$ is $-x + 6$ , not $-x - 6$ : a leading minus flips every sign inside.
Confusing simplifying with evaluating. Combining like terms to get $8x$ is simplifying, no value needed. Substituting a number to reach a single value is evaluating.

Key takeaways

To evaluate, substitute each variable’s value in its own parentheses to protect its sign, then apply the full order of operations: $(-3)^2 = 9$ , while $-3^2 = -9$ .
Like terms have the same variable part (same letters, same powers). To combine them, add or subtract the coefficients and keep the variable part unchanged: $3x + 5x = 8x$ .
Combining like terms is legal because of the distributive property run backward: $3x + 5x = x(3 + 5) = 8x$ . It works only when the variable parts match exactly; sharing just part of a variable part, like the $x$ in $3x$ and $5x^2$ , is not enough.
To simplify an expression with parentheses, distribute first ( $a(b + c) = ab + ac$ , with a leading minus acting as $-1$ ), then collect like terms.
Simplifying rewrites an expression in an equivalent, usually shorter form without substituting any value, so it stays valid for every input (it can even collapse to a plain number, as $x - x$ does to $0$ ); evaluating substitutes a specific value to reach a single number. Simplify once, then evaluate as many times as you need.

Evaluating and Simplifying Expressions

Learning goals

Evaluating with the full order of operations

When the input is negative

Two variables and grouped pieces

Simplifying: the idea of like terms

Simplifying with the distributive property

Evaluating versus simplifying

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Core practice