8.1 Solving One-Variable Linear Equations

Why One-Variable Linear Equations Matter on the GED

Solving one-variable linear equations is the single most tested skill family on the GED Mathematical Reasoning test. It sits inside assessment target A.2 (Write, manipulate, and solve linear equations), part of the Algebraic Problem Solving content area that makes up 55% of the whole exam. But its reach goes further than its own questions: every real-world word problem in the next section, every system of equations later in this chapter, and every inequality in Chapter 9 all reduce to the same core skill — isolating a variable using the properties of equality. If you cannot solve a multi-step linear equation quickly and accurately, every downstream algebra topic on test day becomes harder. This is also one of the few A.2 skills that can appear in the no-calculator section (the first 5 items), so fluency with basic equation-solving steps — without needing a calculator to check your arithmetic — is a real test-day requirement, not just a study convenience.

Core Terms and the Properties of Equality

A linear equation is an equation in which the variable appears only to the first power (no exponents, no variable in a denominator). Solving one means finding the value of the variable that makes the equation true. Every step you take must preserve balance — whatever you do to one side of the equation, you must do to the other side. This rule is formalized in the properties of equality:

A coefficient is the number multiplying a variable (in 5x, 5 is the coefficient). A constant is a term with no variable attached. Like terms share the same variable raised to the same power (3x and 7x are like terms; 3x and 7x² are not) and can be combined by adding or subtracting their coefficients.

The Standard Solving Order

Most GED equations combine several of these ideas in one problem. Use this consistent order every time:

1. Distribute — clear any parentheses using the distributive property: a(b + c) = ab + ac.
2. Clear fractions or decimals — multiply every term on both sides by the least common denominator (LCD), or by a power of 10, so you are working with whole numbers.
3. Combine like terms on each side of the equation separately.
4. Move variable terms to one side using addition or subtraction property of equality.
5. Move constants to the other side.
6. Divide by the coefficient to isolate the variable.
7. Check by substituting your answer back into the original equation.

Worked Example 1 — Distribution and Variables on Both Sides

Solve: 3(x − 4) + 5 = 2x + 7

• Distribute: 3x − 12 + 5 = 2x + 7
• Combine like terms on the left: 3x − 7 = 2x + 7
• Subtract 2x from both sides: x − 7 = 7
• Add 7 to both sides: x = 14
• Check: 3(14 − 4) + 5 = 3(10) + 5 = 35, and 2(14) + 7 = 35. Balanced.

Worked Example 2 — Clearing Fractions

Solve: (2/3)x + 1 = (1/2)x − 2

The LCD of 3 and 2 is 6. Multiply every term by 6:

• 6 × (2/3)x + 6 × 1 = 6 × (1/2)x − 6 × 2
• 4x + 6 = 3x − 12
• Subtract 3x: x + 6 = −12
• Subtract 6: x = −18

A common mistake here is multiplying only one side of the equation by the LCD, or only one term on a side — this destroys the balance and gives a wrong answer. Every single term, on both sides, must be multiplied.

Special Cases: No Solution and Infinitely Many Solutions

Not every linear equation has exactly one numeric answer. Watch for these two outcomes, which the GED tests specifically because test-takers often assume every equation must produce a single number:

If the variable cancels out entirely when you combine like terms, you're looking at one of these two special cases — check whether the remaining numbers are equal (infinite solutions) or unequal (no solution).