Simplify and evaluate algebraic expressions

SECTION 1

Historical Context & Motivation

Have you ever wanted a shortcut to describe a rule that works for any number? That is exactly what algebra gives us. Instead of writing a new problem every time a number changes, we use letters called variables to stand for numbers that can change.

Long ago, people wrote math problems out in full sentences. This was slow and confusing. Over hundreds of years, mathematicians invented symbols to make math faster and clearer, which led to the algebra we use today.

820

The Word "Algebra"

The mathematician al-Khwarizmi wrote a famous book. The word algebra comes from its title.

1591

Letters for Numbers

François Viète began using letters to stand for unknown values, a huge step toward modern algebra.

1637

Modern Symbols

René Descartes started using letters like x, y, and z for unknowns, which we still do today.

Today

Algebra Everywhere

Algebra powers video games, apps, sports stats, and the money math you do every day.

In this lesson, you will learn two key skills: how to simplify an expression to make it shorter, and how to evaluate it by plugging in numbers to find its value.

SECTION 2

Core Principles & Definitions

An algebraic expression is a math phrase that can have numbers, variables (letters), and operations like adding or multiplying. For example, 3x + 5 is an expression. It does not have an equals sign, so it is not an equation.

Variable

A letter that stands for a number that can change, like the x in 3x + 5.

Coefficient

The number multiplied by a variable. In 3x, the coefficient is 3.

Constant

A number by itself that never changes, like the 5 in 3x + 5.

Like Terms

Terms with the exact same variable part, such as 3x and 7x. You can combine them.

To simplify means to combine like terms so the expression is as short as possible. To evaluate means to replace each variable with a given number and then do the math to get one answer.

✦ KEY TAKEAWAY

Think of like terms like fruit in a basket. You can add 3 apples and 7 apples to get 10 apples, but you cannot add 3 apples and 5 bananas into one group. In math, 3x and 7x are both "x-apples," so they combine to 10x. But 3x and 5 are different, so they stay apart.

SECTION 3

Visual Explanation

The picture below breaks the expression 4x + 3 + 2x into its pieces. Notice how the like terms (the ones with an x) get grouped and combined, while the constant stays on its own.

The two cyan x-terms combine into 6x, while the pink constant 3 stays separate. The final simplified expression is 6x + 3.

See how the order of the terms did not matter? Because addition can be done in any order, we can slide the like terms next to each other before combining them.

SECTION 4

How It Works

There are two main rules that make simplifying and evaluating work. The first is combining like terms. The second is the distributive property, which helps you remove parentheses.

COMBINING LIKE TERMS

a·x + b·x = (a + b)·x

Here a and b are coefficients. Just add the coefficients and keep the same variable. Example: 4x + 2x = 6x.

DISTRIBUTIVE PROPERTY

a(b + c) = a·b + a·c

Multiply the outside number by each term inside the parentheses. Example: 3(x + 2) = 3x + 6.

⚠ Watch Your Signs

When you distribute a negative number, the sign of each term inside changes. For example, −2(x − 4) = −2x + 8. The minus times the minus becomes a plus.

To evaluate, you always substitute (plug in) the number first, then follow the order of operations: parentheses, exponents, multiply and divide, then add and subtract.

SECTION 5

Breaking Down the Steps

The chart below shows the full path from a messy expression to a single value. First you simplify, then you evaluate.

Follow the arrows down: distribute, combine like terms, then substitute the number. The final value is 36.

Quick reference for simplifying and evaluating

Action	What You Do
Distribute	Multiply outside number by each inside term
Combine like terms	Add coefficients of matching variables
Substitute	Replace each variable with its given number
Order of operations	Do parentheses and exponents before adding

SECTION 6

Worked Example

Let us evaluate 5(y − 2) + 3y when y = 4. We will simplify first, then plug in the number.

Simplify then Evaluate: 5(y − 2) + 3y, y = 4

Step 1 — Distribute the 5Multiply 5 by both terms inside the parentheses: 5 × y = 5y and 5 × (−2) = −10.

5y − 10 + 3y

Step 2 — Combine like termsThe like terms are 5y and 3y. Add their coefficients: 5 + 3 = 8. The −10 stays as it is.

8y − 10

Step 3 — Substitute y = 4Replace y with 4: 8(4) − 10. Do the multiplication first.

32 − 10

Step 4 — Finish the arithmeticSubtract to find the final value.

22

✦ WHY SIMPLIFY FIRST?

Simplifying before you plug in is like packing your bag before a trip. If you organize first, the actual math is quick and you make fewer mistakes. You could also plug in right away, but the simplified version is much easier to compute.

SECTION 7

Simplify First vs. Plug In First

You actually have two valid ways to evaluate an expression. Both give the correct answer, but each has trade-offs. The table compares them.

Two paths to the same answer

Method	Strength	Weakness
Simplify first	Fewer numbers to compute; less chance of error	Requires knowing simplifying rules
Plug in first	Works even if you forget how to combine terms	More arithmetic; easier to slip up

✦ BOTH ARE CORRECT

It is like taking two different roads to the same store. One might be shorter, but both get you there. Choose the method you feel most confident with, and check your answer by trying the other way.

SECTION 8

Connection to Equations

Once you can simplify and evaluate expressions, you are ready for equations. An equation has an equals sign and asks you to find the value of the variable that makes both sides equal.

Expressions vs. equations

Expression	Equation
`3x + 5`	`3x + 5 = 20`
No equals sign	Has an equals sign
You simplify or evaluate it	You solve it to find x
Answer is another expression or a value	Answer is a specific number for the variable

The simplifying skills you just learned are the first step in solving equations. When you solve, you often simplify each side first, then work to get the variable by itself. Mastering expressions now makes the next chapter much easier.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In the expression 7m + 4, name the coefficient, the variable, and the constant.

PROBLEM 2 — BASIC CALCULATION

Simplify 6a + 2a.

PROBLEM 3 — INTERMEDIATE

Simplify 4(k + 3) + 2k.

PROBLEM 4 — APPLIED

A movie ticket costs $x and popcorn costs $6. A family buys 4 tickets and 2 popcorns. Write and simplify an expression for the total cost, then evaluate it when x = 10.

PROBLEM 5 — CRITICAL THINKING

Evaluate 3(2p − 1) − 4p when p = 5. Then check your answer by simplifying first.

SUMMARY

Summary

An algebraic expression is a math phrase made of variables, coefficients, and constants. To simplify, you combine like terms by adding their coefficients, and you use the distributive property to remove parentheses.

To evaluate, you substitute a given number for each variable and follow the order of operations. Simplifying before you evaluate keeps the numbers small and helps you avoid mistakes. These skills are the foundation for solving equations in the next stage of algebra.