7.1 Writing, Evaluating & Simplifying Linear Expressions

Why This Section Matters

GED Mathematical Reasoning devotes roughly 55% of the test to Algebraic Problem Solving, and Assessment Target A.1 ("Write, evaluate, and compute with expressions and polynomials") is the largest single target in that domain. Per GED Testing Service's official Assessment Guide for Educators, A.1 carries 10 of the Algebraic domain's 32 leaf indicators — more than any other algebra target, since A.2 through A.7 average only 3 to 5 apiece. This section covers the first three (A.1.a–A.1.c), which deal specifically with linear expressions: expressions in which every variable term has an exponent of 1.

Fluency here is tested directly, and it is the prerequisite skill every later algebra target assumes you already have: you cannot solve a linear equation (Target A.2, Chapter 8) without combining like terms and applying the distributive property first, and you cannot write an equation from a word problem without the translation skill taught here. Because evaluating rational-coefficient expressions is exactly the number-sense skill that does not require a machine, several of the GED's first five no-calculator items are drawn straight from this target.

Core Vocabulary

Lock down this vocabulary before working problems — GED item writers use these terms precisely, and confusing them is a common source of lost points.

• A variable is a letter (commonly x, n, or y) standing in for an unknown or changing quantity. A term is a single number, variable, or product of numbers and variables, separated from other terms by + or − signs — in 4x − 7, "4x" and "−7" are the two terms.
• A coefficient is the numeric factor multiplying a variable (the 4 in 4x). GED items frequently use rational coefficients — fractions or decimals such as (3/4)x or 0.6x — not just whole numbers. A constant is a term with no variable attached (the −7 in 4x − 7).
• Like terms share the exact same variable raised to the exact same power (3x and 5x are like terms; 3x and 5x² are not, even though both contain "x").
• An expression has no equals sign and cannot be "solved," only simplified or evaluated; an equation has an equals sign and can be solved for the variable — that skill is Chapter 8.
• A linear expression keeps every variable's exponent at 1, with no x² or higher power — the restriction that separates this section from the polynomial work in Section 7.2.

Simplifying and Expanding Linear Expressions (A.1.a)

Indicator A.1.a asks you to add, subtract, factor, multiply, and expand linear expressions with rational coefficients — four related skills under one indicator.

Combining like terms. Add or subtract only the coefficients of like terms; every other term stays untouched.

• 5x + 3x − 2 = 8x − 2 (combine the x-terms; the constant −2 is unaffected)
• (1/2)x + (3/4)x = (2/4)x + (3/4)x = (5/4)x (find a common denominator first, exactly as you would when adding plain fractions)

Expanding with the distributive property. For any expression a(b + c), distribute: a(b + c) = ab + ac. This single mechanic is the most heavily tested piece of Target A.1.

• 3(2x + 5) = 6x + 15
• −2(4x − 3) = −8x + 6 — distributing a negative number flips the sign of every term inside the parentheses. This is the single biggest trap in this indicator: many test-takers distribute −2 only to the first term, mistakenly producing −8x − 3 instead of −8x + 6.

Factoring (the reverse of distributing). Pull out the greatest common factor (GCF) shared by every term in the expression.

• 8x − 12: the GCF of 8 and 12 is 4, so 8x − 12 = 4(2x − 3)
• 15x + 20: the GCF of 15 and 20 is 5, so 15x + 20 = 5(3x + 4)

Multi-step problems chain these skills together: 5(2x − 3) − (4x − 7) = 10x − 15 − 4x + 7 = 6x − 8, distributing the 5, then the implied −1 across the second parentheses, then combining 10x − 4x and −15 + 7.

Evaluating Linear Expressions (A.1.b)

Indicator A.1.b asks you to substitute an integer for the variable and compute the resulting numeric value. The process: replace every occurrence of the variable with the given number in parentheses, then follow order of operations (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

Evaluate 4x − 7 when x = −3:

4(−3) − 7 = −12 − 7 = −19

Using parentheses around the substituted value is not optional — writing "4 − 3 − 7" instead of "4(−3) − 7" silently drops the negative sign attached to x and is one of the most common wrong-answer generators on this indicator.

If the variable appears more than once in the expression, substitute into every occurrence:

2x − 5x + 9, at x = 4: 2(4) − 5(4) + 9 = 8 − 20 + 9 = −3

Translating Words into Linear Expressions (A.1.c)

Indicator A.1.c is a reading skill wearing an algebra disguise: turning an English phrase into symbols. GED word problems reuse a small, predictable set of key phrases.

The single highest-value trap in this table is reversed subtraction order. "5 less than a number" means x − 5, not 5 − x: the quantity being subtracted is named first in the English phrase but must come second in the expression. Compare that with "5 minus a number," which correctly reverses to 5 − x. GED item writers exploit this confusion regularly, pairing x − 5 and 5 − x as two of the four answer choices for the same word problem.

Multi-operation phrases combine two rules at once: "twice a number, decreased by 3" → 2x − 3 (multiply first, per "twice," then subtract). Real-world example: a taxi charges a flat $3 fee plus $2 per mile, so the cost for m miles is 2m + 3 — the fee is the constant, and the per-mile rate is the coefficient.

Exam Scenario

A GED item states: "A gym charges a $25 sign-up fee plus $15 per month of membership," then asks for an expression for the total cost over m months and asks you to evaluate it at m = 6. Recognizing "$15 per month" as 15m and the "$25 sign-up fee" as the constant gives 15m + 25; substituting m = 6 gives 15(6) + 25 = 115. Expect wrong-answer choices built from swapping the terms (25m + 15) or misplacing the parentheses around the sum instead of only the substituted value.