## AI Linear Equation Solver

### Solving Linear Equations

#### 1. Solving Linear Equations with Arithmetic Operations

**Definition**:

– A linear equation is an equation of the form \( ax + b = c \), where \( x \) is the unknown, and \( a \), \( b \), and \( c \) are constants.

**Basic Steps:**

1. Addition/Subtraction: To isolate the variable, add or subtract the same value from both sides.

2. Multiplication/Division: To solve for the variable, multiply or divide both sides by the same non-zero value.

Example: Solve \( 3x + 5 = 14 \).

Steps:

1. Subtract 5 from both sides: \( 3x + 5 – 5 = 14 – 5 \).

2. Simplify: \( 3x = 9 \).

3. Divide by 3: \( x = \frac{9}{3} \).

4. Simplify: \( x = 3 \).

Solution: \( x = 3 \).

#### 2. Solving Linear Equations with One Unknown

Example: Solve \( 4x – 7 = 9 \).

Steps:

1. Add 7 to both sides: \( 4x – 7 + 7 = 9 + 7 \).

2. Simplify: \( 4x = 16 \).

3. Divide by 4: \( x = \frac{16}{4} \).

4. Simplify: \( x = 4 \).

Solution: \( x = 4 \).

#### 3. Linear Equations Involving Brackets

Example: Solve \( 2(x + 3) = 14 \).

Steps:

1. Expand the brackets: \( 2x + 6 = 14 \).

2. Subtract 6 from both sides: \( 2x + 6 – 6 = 14 – 6 \).

3. Simplify: \( 2x = 8 \).

4. Divide by 2: \( x = \frac{8}{2} \).

5. Simplify: \( x = 4 \).

Solution: \( x = 4 \).

#### 4. Linear Equations Involving Fractions

Example: Solve \( \frac{2x}{3} – \frac{1}{2} = \frac{1}{6} \).

Steps:

1. Find the common denominator (6) and rewrite each fraction:

\[ \frac{2x \cdot 2}{3 \cdot 2} – \frac{1 \cdot 3}{2 \cdot 3} = \frac{1}{6} \]

\[ \frac{4x}{6} – \frac{3}{6} = \frac{1}{6} \]

2. Combine the fractions:

\[ \frac{4x – 3}{6} = \frac{1}{6} \]

3. Multiply both sides by 6 to clear the denominator:

\[ 4x – 3 = 1 \]

4. Add 3 to both sides:

\[ 4x – 3 + 3 = 1 + 3 \]

\[ 4x = 4 \]

5. Divide by 4:

\[ x = \frac{4}{4} \]

\[ x = 1 \]

Solution: \( x = 1 \).

#### Summary of Steps for Solving Linear Equations:

**1. Simplify each side of the equation:**

- Combine like terms.
- Remove parentheses by distributing.

**2. Isolate the variable term:**

- Use addition or subtraction to move constants to the other side of the equation.

**3. Solve for the variable:**

- Use multiplication or division to isolate the variable.

**4. Check your solution:**

- Substitute the solution back into the original equation to verify.