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  1. GED Math

  3. Write and Evaluate Algebraic Expressions

GED MATHEMATICAL REASONING • ALGEBRAIC PROBLEM SOLVING

Write and Evaluate Algebraic Expressions

Turn real-world situations into the language of algebra, then calculate their value.

SECTION 1

Where Did Algebra Come From?

Long before modern textbooks existed, people needed ways to solve practical problems — dividing land, splitting wages, or calculating trade goods. Ancient civilizations from Babylon to Egypt developed methods for working with unknown quantities, but they described everything in words and sentences. The leap to using symbols and letters to represent unknowns changed mathematics forever. Understanding this history can help you see that algebra is not an abstract puzzle — it is a tool that humans built to solve everyday problems, just like the ones you will encounter on the GED.

1800 BCE
Babylonian Word Problems
Clay tablets show Babylonians solving problems about area and trade using step-by-step word instructions — essentially algebra without symbols.
820 CE
Al-Khwarizmi's 'Al-Jabr'
The Persian mathematician al-Khwarizmi wrote the book that gave algebra its name. He described systematic methods for solving equations using words.
1637
Descartes Introduces x, y, z
René Descartes popularized using letters like x, y, and z for unknowns and a, b, c for known values — the notation we still use today.
Today
Algebra in Daily Life
From calculating your paycheck to comparing cell phone plans, algebraic expressions appear constantly. The GED tests your ability to write and evaluate these expressions in real-world contexts.

The central question this lesson answers is straightforward: how do you take a situation described in English — "a worker earns $15 per hour plus a $50 bonus" — and rewrite it as an algebraic expression that you can then evaluate for any number of hours? Mastering this skill is essential because roughly 55% of the GED Math test focuses on algebraic problem solving, and writing and evaluating expressions is the foundation for all of it.

SECTION 2

Core Principles and Definitions

Before you can write or evaluate any algebraic expression, you need a clear understanding of the vocabulary. These are the building blocks you will use throughout this lesson and on the GED exam. Take a moment to study each term below — they will come up again and again.

1

Variable

A letter (such as x, n, or t) that represents an unknown or changing quantity. Think of it as a blank space that can hold different numbers.
2

Constant

A fixed number that does not change. In the expression 3x + 7, the number 7 is a constant.
3

Coefficient

The number multiplied by a variable. In 3x, the coefficient is 3. It tells you how many groups of the variable you have.
4

Term

A single piece of an expression separated by addition or subtraction. The expression 4x + 2y − 5 has three terms: 4x, 2y, and −5.
5

Expression vs. Equation

An expression has no equal sign — it is a phrase, not a complete sentence. An equation uses = to show two expressions are equal. This lesson focuses on expressions.
✦ KEY TAKEAWAY
Think of an algebraic expression like a recipe. The variables are the ingredients you can change (like the number of servings), the coefficients tell you how much of each ingredient per serving, and the constants are the fixed amounts that stay the same no matter what. Evaluating the expression is like calculating the total ingredients once you know how many servings you are making.
SECTION 3

Anatomy of an Algebraic Expression

The diagram below breaks down the expression 3x + 2y − 5 into its individual parts. Study how each component — variable, coefficient, constant, and operator — fits together. Being able to identify these parts quickly is the first step toward both writing and evaluating expressions.

ANATOMY OF AN ALGEBRAIC EXPRESSION3x+2y−5COEFFICIENTVARIABLECOEFFICIENTVARIABLECONSTANT+ and − are OPERATORS that separate TERMSTerm 1: 3xcoefficient 3 × variable xTerm 2: 2ycoefficient 2 × variable yTerm 3: −5constant (no variable)
The expression 3x + 2y − 5 contains three terms. The first two terms each have a coefficient and a variable, while the third term is a constant. Operators (+ and −) separate the terms.

Notice that the expression has three terms separated by the addition and subtraction operators. The term 3x tells you to multiply 3 by whatever value x represents. The term 2y does the same with y. The constant −5 is subtracted regardless of the values of x or y. When the GED asks you to identify parts of an expression, this is exactly what they mean.

SECTION 4

Translating Words into Algebra

One of the most tested skills on the GED is translating a verbal phrase into an algebraic expression. The key is recognizing which English words correspond to which mathematical operations. Once you learn these translations, writing expressions becomes almost automatic.

Common verbal-to-algebraic translations you will need on the GED.
OperationKey Words / PhrasesExample in WordsAlgebraic Expression
Additionsum, plus, more than, increased by, totala number increased by 9x + 9
Subtractiondifference, minus, less than, decreased by, fewer7 less than a numberx − 7
Multiplicationproduct, times, of, per, each, twice, double, tripletwice a number2x
Divisionquotient, divided by, ratio, split equally, per (in some contexts)a number divided by 4x / 4
Exponentsquared, cubed, to the power ofa number squaredx²
⚠️ Watch Out: "Less Than" Reverses Order
The phrase "7 less than a number" becomes x − 7, not 7 − x. The words "less than" and "subtracted from" flip the order from how you read them in English. This is one of the most common mistakes on the GED — pay close attention to the order.

Evaluating an Expression: Substitution

Once you have an algebraic expression, evaluating it means plugging in a specific number for each variable and then simplifying using the order of operations. You can remember the order of operations with the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

EVALUATION PROCESS
If E = 3x + 2y − 5, and x = 4, y = 3, then E = 3(4) + 2(3) − 5 = 12 + 6 − 5 = 13
Replace each variable with the given value, then follow the order of operations to simplify to a single number.
ORDER OF OPERATIONS (PEMDAS)
P → E → M/D → A/S
P = Parentheses first, E = Exponents, M/D = Multiply or Divide left to right, A/S = Add or Subtract left to right. The GED formula sheet does not list this, so commit it to memory.
SECTION 5

From Situation to Expression — A Visual Guide

The GED frequently presents word problems that ask you to write an expression before evaluating it. The diagram below illustrates the complete process: start with a real-world scenario, identify the variable, translate each piece into algebra, and then combine the pieces into a single expression. Follow the arrows to trace how an everyday situation becomes math.

FROM SITUATION TO EXPRESSIONREAL-WORLD SCENARIO"A plumber charges $40 per hour plus a $75 service fee."STEP 1: IDENTIFY THE VARIABLELet h = number of hours workedSTEP 2A: TRANSLATE RATE$40 per hour → 40hSTEP 2B: TRANSLATE FIXED FEE$75 fee → + 75STEP 3: COMBINE INTO ONE EXPRESSIONTotal Cost = 40h + 75To evaluate: substitute h = 3 → 40(3) + 75 = 120 + 75 = 195 → The plumber charges $195
This flowchart shows the three-step process: (1) identify the variable, (2) translate each piece of the scenario, and (3) combine the pieces into a single algebraic expression. The bottom bar shows how to evaluate the expression for a specific value.

Notice that the key phrase "per hour" signals multiplication — the rate of $40 is multiplied by the variable h. The word "plus" signals addition for the fixed $75 service fee. On the GED, you may also see problems with subtraction ("discount of"), division ("split evenly among"), or exponents ("squared"). Each follows the same process: identify the variable, translate each part, and combine.

SECTION 6

Worked Example

Let's work through a full GED-style problem from start to finish. Pay close attention to each step — this is the exact process you will use on test day.

📝 Problem
Maria works at a coffee shop. She earns $12 per hour and gets a $5 tip bonus each shift. Write an expression for her total earnings in a single shift, then evaluate it if she works 6 hours.

Solution: Maria's Earnings

Step 1 — Identify the Variable

The quantity that changes from shift to shift is the number of hours Maria works. Let h represent the number of hours she works in one shift.
h = number of hours

Step 2 — Translate the Hourly Rate

"$12 per hour" means she earns $12 for each hour. The word "per" signals multiplication. So her hourly earnings are represented by 12h.
12h

Step 3 — Add the Constant

The $5 tip bonus is a fixed amount added to every shift, regardless of hours. This is a constant. We add it to the hourly earnings.
Total earnings = 12h + 5

Step 4 — Evaluate for h = 6

Substitute 6 for h: 12(6) + 5. Following the order of operations, multiply first: 12 × 6 = 72. Then add: 72 + 5 = 77.
Maria earns $77 for a 6-hour shift.
✓ CHECK YOUR WORK
After evaluating, do a quick reasonableness check. Maria earns $12 per hour, so 6 hours should give her about $72 from wages alone. Adding a $5 tip makes $77. That makes sense — it is close to our estimate. If your answer had been $770 or $7, you would know something went wrong with the arithmetic.
SECTION 7

Common Pitfalls and How to Avoid Them

Many GED test-takers lose points not because the algebra is hard, but because of small, preventable mistakes. The table below lists the most common errors along with the correct approach. Reviewing these before test day can be worth several extra points.

Review these common pitfalls before test day.
Common MistakeWhy It's WrongCorrect Approach
Writing "5 less than x" as 5 − x"Less than" reverses the order in English. You are taking 5 away from x, not x away from 5.x − 5
Ignoring order of operationsEvaluating 3 + 4 × 2 as 14 instead of 11 — addition was done before multiplication.Always multiply/divide before adding/subtracting: 3 + 8 = 11
Forgetting parentheses around negative substitutionsSubstituting x = −3 into 2x² as 2 × −3² = −18 instead of 2(−3)² = 18.Always wrap negative values in parentheses: 2(−3)² = 2(9) = 18
Confusing "twice a number plus 3" with "twice the sum of a number and 3"The first is 2x + 3; the second is 2(x + 3). These give different results.Look for words like "the sum of" or "the quantity" — they signal parentheses are needed: 2(x + 3)
✦ KEY TAKEAWAY
The most dangerous mistakes are the small ones: reversed subtraction order, skipped parentheses, and misapplied order of operations. On the GED, the wrong answer choices (distractors) are specifically designed to match these common errors. If your answer matches a choice but you feel unsure, go back and double-check these three areas.
SECTION 8

From Expressions to Equations and Beyond

Writing and evaluating expressions is the foundation, but the GED builds on this skill in important ways. Once you are comfortable with expressions, you will move on to equations (which include an equal sign) and inequalities (which use symbols like < and >). The table below shows how expressions connect to these more advanced topics.

How algebraic expressions connect to equations, inequalities, and functions on the GED.
ConceptWhat It Looks LikeKey Difference from Expressions
Expression3x + 7No equal sign. You evaluate it by substituting a value.
Equation3x + 7 = 22Has an equal sign. You solve for x by isolating the variable.
Inequality3x + 7 > 22Uses < or > instead of =. Solving gives a range of values.
Functionf(x) = 3x + 7Names the expression so you can reuse it. f(x) is read as "f of x."

Every equation, inequality, and function you will see on the GED is built from algebraic expressions. If you can write and evaluate expressions with confidence, you already have the core skill needed for over half the test. In the next lesson, you will learn how to take an expression like 3x + 7, set it equal to a value, and solve for the unknown variable.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL
A gym membership costs a one-time sign-up fee of $50 plus $30 per month. Which of the following algebraic expressions represents the total cost after m months?
PROBLEM 2 — BASIC CALCULATION
Evaluate the expression 5x − 3 when x = 7.
PROBLEM 3 — INTERMEDIATE
A cell phone plan charges $0.10 per text message and $0.05 per minute of talk time, plus a flat monthly fee of $25. If t represents the number of text messages and m represents the number of talk minutes, which expression gives the total monthly bill?
PROBLEM 4 — APPLIED
Jamal works part-time while attending community college and wants to buy a laptop for his classes. The laptop costs $800. He already has $200 saved from previous paychecks and plans to set aside $50 from each weekly paycheck. He writes the expression 200 + 50w to represent his total savings after w weeks. After how many full weeks of saving will his total savings first equal or exceed $800? Enter the number of weeks as a whole number.
PROBLEM 5 — CRITICAL THINKING
Two students are asked to write an expression for "three times the sum of a number and 4." Student A writes 3n + 4. Student B writes 3(n + 4). Which student wrote the correct expression?
SUMMARY

Lesson Summary

An algebraic expression is a mathematical phrase containing variables (letters representing unknown values), coefficients (numbers multiplied by variables), constants (fixed numbers), and operations. To write an expression from a word problem, identify the variable, translate key phrases — "per" means multiply, "more than" means add, "less than" means subtract (in reverse order) — and combine all the pieces. Always watch for phrases like "the sum of" that signal parentheses are needed.

To evaluate an expression, substitute the given number for each variable, then simplify using PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). Use parentheses when substituting negative numbers to avoid sign errors. This skill forms the foundation for solving equations, inequalities, and working with functions — topics that together make up over half of the GED Mathematical Reasoning test.

Varsity Tutors • GED Mathematical Reasoning • Write and Evaluate Algebraic Expressions