Solving Linear Equations Step by Step

Linear equations are one of the first places algebra starts to feel abstract, but the method for solving them is consistent across almost every version you'll encounter. This walkthrough covers four common equation types, with every answer verified by substituting it back into the original equation, so you can see exactly why each step works.

The core idea: keep both sides balanced

A linear equation is a statement that two expressions are equal. Whatever you do to one side, you must do to the other, or the equation stops being true. The goal in solving is always the same: isolate the variable on one side by undoing operations in reverse order.

Example 1: 3x + 5 = 20

Step 1: Subtract 5 from both sides to undo the addition. 3x + 5 - 5 = 20 - 5, which simplifies to 3x = 15.

Step 2: Divide both sides by 3 to undo the multiplication. 3x / 3 = 15 / 3, which gives x = 5.

Check: Substitute x = 5 back into the original equation. 3(5) + 5 = 15 + 5 = 20. This matches the right side, confirming the answer.

The order matters here: addition and subtraction are undone before multiplication and division, working from the outside in toward the variable.

Example 2: 2(x - 4) = 10

This one has parentheses, which changes the first step.

Step 1: Divide both sides by 2 first, since 2 multiplies the entire parenthetical expression. 2(x - 4) / 2 = 10 / 2, which gives x - 4 = 5.

Step 2: Add 4 to both sides. x - 4 + 4 = 5 + 4, which gives x = 9.

Check: Substitute x = 9. 2(9 - 4) = 2(5) = 10. This matches the right side.

An alternative approach is to distribute the 2 across the parentheses first, turning the left side into 2x - 8 = 10, then solving from there. Both methods lead to the same answer; dividing first is often faster when the number outside the parentheses divides evenly into the number on the other side.

Example 3: 5x - 3 = 2x + 9

This equation has the variable on both sides, which means you need an extra step before isolating x.

Step 1: Subtract 2x from both sides to bring all the x terms to one side. 5x - 2x - 3 = 2x - 2x + 9, which simplifies to 3x - 3 = 9.

Step 2: Add 3 to both sides. 3x - 3 + 3 = 9 + 3, which gives 3x = 12.

Step 3: Divide both sides by 3. 3x / 3 = 12 / 3, which gives x = 4.

Check: Substitute x = 4 into both sides of the original equation. Left side: 5(4) - 3 = 20 - 3 = 17. Right side: 2(4) + 9 = 8 + 9 = 17. Both sides equal 17, confirming the answer.

Example 4: x/4 + 1 = 6

Step 1: Subtract 1 from both sides. x/4 + 1 - 1 = 6 - 1, which gives x/4 = 5.

Step 2: Multiply both sides by 4 to undo the division. (x/4) × 4 = 5 × 4, which gives x = 20.

Check: Substitute x = 20. 20/4 + 1 = 5 + 1 = 6. This matches the right side.

A common mistake to watch for

A frequent error in Example 3's type of problem is subtracting 2x from only one side of the equation, which breaks the balance and leads to a wrong answer. Another common mistake in Example 2's type of problem is forgetting to apply the division or distribution to every term inside the parentheses, not just the first one. Writing out each step explicitly, rather than trying to combine steps mentally, is the most reliable way to avoid both of these errors.

Why checking your answer matters

Substituting your answer back into the original equation, before distributing or combining anything, is the fastest way to catch a mistake. If both sides don't come out equal, you know a step went wrong somewhere, and you can retrace your work rather than submitting an unchecked answer. This habit matters more as equations get longer, since a single sign error early on can otherwise go unnoticed until much later in a multi-step problem.

According to the Common Core State Standards Initiative's Grade 8 mathematics overview, analyzing and solving linear equations is one of the central skills students build in that grade, specifically listed as part of the "Expressions and Equations" content area, which reflects how foundational this step-by-step process is to algebra more broadly.

This is a precision habit, not just an answer habit

The Common Core State Standards Initiative's Standards for Mathematical Practice list "attend to precision" as one of the general practices expected across all grade levels of math, not just algebra specifically. Writing out each step of a linear equation explicitly, rather than skipping steps mentally, is a direct application of that practice: it forces you to track exactly what you did to both sides of the equation, which is also what makes the final substitution check meaningful rather than just a formality.

Key takeaways

• Solve a linear equation by undoing operations in reverse order, addition and subtraction first, then multiplication and division.
• When an equation has parentheses, you can divide through first if the outside number divides evenly, or distribute first; both methods give the same answer.
• When the variable appears on both sides, combine the variable terms onto one side before isolating it.
• Always check your answer by substituting it back into the original equation, not a simplified version of it.
• The most common mistakes come from applying an operation to only part of an equation, not the whole thing.