Solve Linear Equations — Solve single-variable linear equations.

The ability to solve a linear equation is one of the oldest and most practical skills in mathematics. Long before anyone wrote equations with letters and symbols, ancient engineers, builders, and tradespeople faced a fundamental problem: if you know the total and some of the parts, how do you find what is missing? Whether an Egyptian surveyor was calculating the length of a canal or a Babylonian merchant was figuring out the price per unit of grain, the underlying logic was the same—work backward from what you know to find the unknown quantity. This same logic is precisely what you use on the job when you rearrange Ohm's Law to find current, or back-calculate the length of conduit needed for a run.

The central question that linear equations answer is deceptively simple: What value of the unknown makes both sides of the equation equal? Every technique you will learn in this lesson—combining like terms, using inverse operations, clearing fractions—is a tool for answering that question efficiently and accurately. Mastering these techniques is not optional for the IBEW Electrical Training Alliance Aptitude Test; it is the single most tested algebraic skill.

Before diving into solving procedures, you need a clear understanding of the building blocks. A linear equation in one variable is any equation that can be written in the form ax + b = c, where x is the unknown, and a, b, and c are known numbers (with a ≠ 0). The word "linear" means the variable appears only to the first power—no x², no √x, no 1/x. These equations graph as straight lines, hence the name, and they always have exactly one solution (assuming a ≠ 0).

The most powerful mental model for solving linear equations is the balance scale. The equals sign is the fulcrum; whatever is on the left side must weigh the same as what is on the right side. The diagram below illustrates how solving 2x + 3 = 11 works step by step: each operation you perform is applied to both sides simultaneously, keeping the scale level until x stands alone.

Notice that at every stage the scale remains balanced. This is the inviolable rule of equation solving: perform the same operation on both sides. If you subtract 3 from the left but forget to subtract 3 from the right, the scale tips and you get a wrong answer. This is the most common source of errors on the aptitude test—always double-check that your operation was applied to the entire right-hand side, not just one term.

Every single-variable linear equation, no matter how complex it initially appears, can be reduced to the standard form ax + b = c, and then solved in two moves. The general solution procedure relies on two inverse operations applied in the correct order. Understanding the underlying formulas ensures you can handle any variation the test throws at you.

When the equation is more complex—variables on both sides, parentheses, or fractions—additional preliminary steps are required before you reach standard form. The general algorithm expands to the following sequence.

1. Clear fractions or decimals by multiplying every term on both sides by the least common denominator (LCD).
2. Distribute to eliminate parentheses using the distributive property: a(b + c) = ab + ac.
3. Combine like terms on each side of the equation separately.
4. Move variable terms to one side and constants to the other using addition or subtraction.
5. Divide both sides by the coefficient to solve for x.
6. Check by substituting the answer back into the original equation.

Not all linear equations look the same on the test. Recognizing the type of equation you are dealing with lets you choose the most efficient approach. The diagram below categorizes the four main types you will encounter, ranging from basic one-step equations to multi-step equations with fractions and variables on both sides.

Let us work through a multi-step equation of the type you will encounter on the IBEW aptitude test. This example combines distribution, combining like terms, and variable collection.

Understanding the correct procedure is half the battle; the other half is avoiding the mistakes that even experienced test-takers make under time pressure. The table below identifies the most frequent errors on linear equation problems and pairs each with a concrete strategy to prevent it.

Single-variable linear equations are the gateway to more complex algebraic problem solving. Once you have mastered the technique of isolating a variable through inverse operations, you can apply the exact same logic to rearranging multi-variable formulas—something you will do constantly in the electrical trades. Ohm's Law (V = IR), the power formula (P = IV), and voltage drop calculations (V_d = 2 × K × I × D / CM) are all linear equations in disguise. Solving for any one variable requires the same skills practiced in this lesson.

On the IBEW aptitude test, the algebra section tests whether you can reliably isolate a variable, handle sign changes, and work with fractions and decimals. These are the exact skills that apprenticeship instructors build upon when you begin calculating wire sizes, conduit fill, and load balancing. Invest in these fundamentals now and every formula you encounter later will feel like a variation of something you already know.