Introduction
Linear equations in one variable form the foundation of algebra. These are equations that contain only one unknown (usually denoted as x) and the highest power of the variable is 1.
This pattern is important because it helps in developing basic problem-solving skills and logical reasoning - essential for all higher-level math topics.
Pattern: Linear Equations in One Variable
Pattern
The key concept: Simplify both sides and isolate the variable to find its value.
The general form of a linear equation in one variable is:
ax + b = c
To solve, bring all variable terms to one side and constants to the other side, then divide by the coefficient of x.
Step-by-Step Example
Question
Solve for x: 3x + 5 = 20
Solution
Step 1: Write the given equation
3x + 5 = 20.Step 2: Move the constant to the other side
Subtract 5 from both sides:
⇒ 3x = 20 - 5 = 15.Step 3: Isolate x by dividing
Divide both sides by 3:
⇒ x = 15 ÷ 3 = 5.Final Answer:
5Quick Check:
Substitute x = 5 into the original equation: 3(5) + 5 = 15 + 5 = 20 ✅
Quick Variations
1. Variables on both sides → e.g., 2x + 3 = x + 7
2. Fractional coefficients → e.g., (x/2) + 3 = 5
3. Negative or decimal coefficients → e.g., -0.5x + 4 = 2
Trick to Always Use
- Step 1: Collect all variable terms on one side.
- Step 2: Move constants to the other side.
- Step 3: Simplify and divide to get the final answer.
Summary
Summary
In the Linear Equations in One Variable pattern:
- Keep the equation balanced - whatever you do to one side, do to the other.
- Always simplify step by step.
- Substitute back to verify your solution quickly.
Hint: Subtract the constant first, then divide by the coefficient.
Common Mistakes: Forgetting to subtract the constant before dividing; arithmetic errors.