Write and Evaluate Algebraic Expressions - GED Math

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.

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.

Constant

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

Coefficient

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

Term

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

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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.

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.

Operation	Key Words / Phrases	Example in Words	Algebraic Expression
Addition	sum, plus, more than, increased by, total	a number increased by 9	`x + 9`
Subtraction	difference, minus, less than, decreased by, fewer	7 less than a number	`x − 7`
Multiplication	product, times, of, per, each, twice, double, triple	twice a number	`2x`
Division	quotient, divided by, ratio, split equally, per (in some contexts)	a number divided by 4	`x / 4`
Exponent	squared, cubed, to the power of	a number squared	`x²`

Common verbal-to-algebraic translations you will need on the GED.

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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.

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.

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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

1
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
2
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
3
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
4
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.

Common Mistake	Why It's Wrong	Correct 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 operations	Evaluating 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 substitutions	Substituting 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)

Review these common pitfalls before test day.

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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.

Concept	What It Looks Like	Key Difference from Expressions
Expression	`3x + 7`	No equal sign. You evaluate it by substituting a value.
Equation	`3x + 7 = 22`	Has an equal sign. You solve for x by isolating the variable.
Inequality	`3x + 7 > 22`	Uses < or > instead of =. Solving gives a range of values.
Function	`f(x) = 3x + 7`	Names the expression so you can reuse it. f(x) is read as "f of x."

How algebraic expressions connect to equations, inequalities, and functions on the GED.

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

Question 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?

A. 50m + 30
B. 30m + 50
C. 80m
D. 30 + 50 + m

Show answer (B)

The correct answer is B: 30m + 50. The monthly fee of $30 is multiplied by the number of months m, giving 30m. The one-time sign-up fee of $50 is a constant added to that, giving 30m + 50. Option A (50m + 30) incorrectly treats $50 as the per-month charge and $30 as the fixed fee — that reverses the two values. Option C (80m) incorrectly multiplies the total of both numbers by m, which would mean you pay the sign-up fee every month. Option D (30 + 50 + m) simply adds m rather than multiplying it by the rate.

Question 2: BASIC CALCULATION

Evaluate the expression 5x − 3 when x = 7.

A. 38
B. 32
C. 14
D. 10

Show answer (B)

The correct answer is B: 32. Substitute 7 for x: 5(7) − 3 = 35 − 3 = 32. Option A (38) results from adding 3 instead of subtracting: 5(7) + 3 = 35 + 3 = 38. Option C (14) could result from incorrectly computing 7 × 2 = 14, for example by misreading the coefficient 5 as 2 or by misunderstanding the expression structure and calculating (x + x) = 7 + 7 = 14. Option D (10) would come from computing 5 + 7 − 3 + 1 = 10, reflecting a misunderstanding of what the coefficient means — treating multiplication as addition.

Question 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?

A. 0.10t + 0.05m + 25
B. 0.10m + 0.05t + 25
C. 0.15(t + m) + 25
D. 25t + 0.10m + 0.05

Show answer (A)

The correct answer is A: 0.10t + 0.05m + 25. Text messages cost $0.10 each, so the text cost is 0.10 × t. Talk minutes cost $0.05 each, so the talk cost is 0.05 × m. The flat fee of $25 is added as a constant. Option B swaps the rates — it charges the text rate for minutes and vice versa. Option C incorrectly combines the two different rates into one rate of $0.15 applied to the total of texts and minutes, which only works if both cost the same amount. Option D treats $25 as a per-text charge and places the other rates incorrectly.

Question 4: APPLIED

2 pts

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.

Show answer

Point 1 (1 pt): The student correctly sets up the evaluation by substituting into the expression and solving the inequality: 200 + 50w ≥ 800, which simplifies to 50w ≥ 600, so w ≥ 12. Point 2 (1 pt): The student provides the correct final answer: 12 weeks. Verification: At w = 12, the expression gives 200 + 50(12) = 200 + 600 = 800, which equals the laptop cost. Any answer of 12 receives full credit. Partial credit (1 pt) may be awarded if the student shows a correct setup (such as writing 200 + 50w = 800 or 50w = 600) but makes an arithmetic error in the final calculation.

Question 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?

A. Student A is correct because you multiply 3 by n first, then add 4.
B. Student B is correct because "the sum of a number and 4" must be grouped in parentheses before multiplying by 3.
C. Both students are correct because 3n + 4 and 3(n + 4) always give the same result.
D. Neither student is correct because the expression should be written as (3n)(4).

Show answer (B)

The correct answer is B. The phrase "the sum of a number and 4" describes a single quantity: n + 4. The word "times" then tells us to multiply that entire sum by 3, giving 3(n + 4). Student A's expression, 3n + 4, means "three times a number, then add 4" — which is a different phrase entirely. To see why they differ, try n = 2: Student A gets 3(2) + 4 = 6 + 4 = 10, while Student B gets 3(2 + 4) = 3(6) = 18. Since the results differ, the expressions are not equivalent, which eliminates Option C. Option D proposes (3n)(4) = 12n, which means "twelve times a number" and does not match the original phrase at all.

SUMMARY

Lesson Summary

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.