Solving Linear Equations: Inverse Operations and Steps
Gianluca
December 19, 2025
A Linear Equation is an equation for a straight line. Solving it means finding the value of the variable (usually \(x\)) that makes the equation true.
Core Principles
Inverse Operations: To solve for \(x\), we “undo” operations to isolate the variable.
- Addition \(\leftrightarrow\) Subtraction
- Multiplication \(\leftrightarrow\) Division
The Golden Rule: Whatever you do to one side of the equation, you must do to the other side to maintain equality.
Step-by-Step Example
Solve for \(x\):
$$3(x - 4) + 2 = 5x - 2$$Step 1: Distribute Expand the parentheses on the left side.
$$3x - 12 + 2 = 5x - 2$$Step 2: Combine Like Terms Simplify the left side.
$$3x - 10 = 5x - 2$$Step 3: Move variables to one side Subtract \(3x\) from both sides.
$$-10 = 2x - 2$$Step 4: Isolate the variable term Add \(2\) to both sides.
$$-8 = 2x$$Step 5: Solve for \(x\) Divide by \(2\).
$$x = -4$$Step 6: Check your answer Substitute \(-4\) back into the original equation: \(3(-4 - 4) + 2 = 3(-8) + 2 = -24 + 2 = -22\) \(5(-4) - 2 = -20 - 2 = -22\) Both sides match.
Solution: \(x = -4\)
The most frequent mistake isn’t the math—it’s the “moving.” Many students treat the equal sign ($=$) like a door they can just walk through. They write $x + 5 = 12 \Rightarrow x = 12 + 5$.
Remember: Every time a term crosses the equal sign, it must pay a “Border Tax” by changing its sign.
- Wrong: $x + 8 = 10 \Rightarrow x = 10 + 8$ (Result: $18$)
- Correct: $x + 8 = 10 \Rightarrow x = 10 - 8$ (Result: $2$)
This also applies to negative coefficients. If you have $-3x = 12$, you divide by $-3$, you don’t change the $-3$ to $+3$ because the operation is division, not addition.
If you keep getting the wrong results, double-check your signs before doing any calculation. If this still feels tricky, review The Golden Rule of Inverse Operations.
Why reading isn’t enough
Solving linear equations is the foundation for all higher-level math. While the concept is simple, multi-step equations with fractions or parentheses can be tricky.
Weekzen provides a focused environment to practice solving linear equations. We generate problems that target your specific weak points, whether it’s distributing negatives or handling fractions.