Solving Single-Variable Linear Equations

Linear equations in one variable are fundamental in algebra. They help us understand relationships between numbers and appear in many real-world applications. This lesson will cover the basics of solving these equations, step-by-step methods, and practice problems.

1. Definition of a Linear Equation

A linear equation in one variable is an equation that can be written in the form:

where:

• $a$ and $b$ are real numbers,
• $a\ne 0$,
• $x$ is the variable.

The solution to a linear equation is the value of $x$ that makes the equation true.

Examples:

• $2x+3=7$
• $-5x+10=0$
• $\frac{1}{2}x-4=3$

2. Solving Linear Equations Step by Step

The goal is to isolate $x$ on one side of the equation. Here’s a structured approach:

Step 1: Simplify Both Sides

• Distribute if necessary.
• Combine like terms on each side.

Step 2: Move Variables to One Side

• If $x$ appears on both sides, move them to one side by adding or subtracting.

Step 3: Move Constants to the Other Side

• Use addition or subtraction to isolate the term with $x$.

Step 4: Solve for $x$

• Divide or multiply both sides by the coefficient of $x$.

3. Example Solutions

Example 1: Solving a Simple Equation

Solve $3x+5=11$.

Step 1: Subtract 5 from both sides
$$
3x + 5 - 5 = 11 - 5
$$
$$
3x = 6
$$

Step 2: Divide both sides by 3
$$
x = \frac{6}{3}
$$
$$
x = 2
$$

Example 2: Solving an Equation with Parentheses

Solve $2(x-3)=8$.

Step 1: Distribute the 2
$$
2x - 6 = 8
$$

Step 2: Add 6 to both sides
$$
2x = 14
$$

Step 3: Divide by 2
$$
x = \frac{14}{2}
$$
$$
x = 7
$$

Example 3: Solving an Equation with Fractions

Solve $\frac{x}{3}+4=6$.

Step 1: Subtract 4 from both sides
$$
\frac{x}{3} = 2
$$

Step 2: Multiply both sides by 3
$$
x = 6
$$

4. Special Cases

No Solution

If an equation simplifies to a false statement, there is no solution.

Example:
$$
3(x + 2) = 3x + 5
$$
Expanding:
$$
3x + 6 = 3x + 5
$$
Subtracting $3x$ from both sides:
$$
6 = 5
$$
This is false, so there is no solution.

Infinite Solutions

If an equation simplifies to a true statement, there are infinitely many solutions.

Example:
$$
4(x - 1) = 4x - 4
$$
Expanding:
$$
4x - 4 = 4x - 4
$$
Subtracting $4x$ from both sides:
$$
-4 = -4
$$
This is always true, so there are infinitely many solutions.

5. Practice Problems

Solve the following equations:

1.

Try to solve each equation and check your solutions!