Solving Linear Equations in One Variable

Linear Equations in One Variable

Definition and Structure

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

• $ax + b = c$

• Where x is the variable, and a, b, and c are real numbers.

The goal is to find the value of x that makes the equation true. Linear equations in one variable may have one solution, no solution, or infinitely many solutions.

Properties of Equality

To solve linear equations, we use the properties of equality:

• Whatever operation is performed on one side of the equation must also be performed on the other side to maintain balance.

• Common operations include addition, subtraction, multiplication, and division.

Examples of Linear Equations

• $3x = -15$

• $7 - y = 3y$

• $4n - 9n + 6 = 0$

• $z = -2$

Example 1: Solving a Simple Linear Equation

Solve for $x$:

• $2x + 5 = 9$

• Subtract 5 from both sides: $2x = 4$

• Divide both sides by 2: $x = 2$

Example 2: Solving with Decimals

Solve for $c$:

• $0.6 = 2 - 3.5c$

• Subtract 2 from both sides: $0.6 - 2 = -3.5c$

• $-1.4 = -3.5c$

• Divide both sides by $-3.5$: $c = \frac{-1.4}{-3.5} = 0.4$

Combining Like Terms

Definition of Like Terms

Like terms are terms that have the same variable raised to the same power. Combining like terms simplifies equations and makes them easier to solve.

• Example: $-4x - 1 + 5x = 9x + 3 - 7x$

• Combine like terms: $(-4x + 5x) = x$, $(9x - 7x) = 2x$

• Equation becomes: $x - 1 = 2x + 3$

Using the Distributive Property

If an equation contains parentheses, use the distributive property to remove them:

• Example: $2(x - 3) = 5x - 9$

• Apply distributive property: $2x - 6 = 5x - 9$

Solving Equations with Fractions or Decimals

Clearing Fractions

To solve equations with fractions, multiply both sides by the least common denominator (LCD) to eliminate fractions.

• Example: $\frac{y}{3} - \frac{y}{4} = 16$

• LCD of 3 and 4 is 12. Multiply both sides by 12:

• $12 \left( \frac{y}{3} - \frac{y}{4} \right ) = 12 \times 16$

• $4y - 3y = 192$

• $y = 192$

Step-by-Step Method for Solving Linear Equations

1. Clear fractions by multiplying both sides by the LCD.

2. Use the distributive property to remove parentheses.

3. Combine like terms on each side.

4. Use the addition property of equality to get variable terms on one side and constants on the other.

5. Use the multiplication property of equality to isolate the variable.

6. Check the solution in the original equation.

Example 6: Solving a More Complex Equation

• $x + 52 + 12 = 2x - x - 38$

• Combine like terms: $x + 64 = x - 38$

• Subtract $x$ from both sides: $64 = -38$ (No solution)

• Additional info: If the variable terms cancel and the resulting statement is false, the equation has no solution.

Example 7: Solving with Decimals

• $0.3x + 0.1 = 0.27x - 0.02$

• Subtract $0.27x$ from both sides: $0.03x + 0.1 = -0.02$

• Subtract $0.1$ from both sides: $0.03x = -0.12$

• Divide by $0.03$: $x = -4$

Example 8: Common Mistakes

• Given: $3x - 5 = 16$

• Add 5 to both sides: $3x = 21$

• Divide by 3: $x = 7$

• Incorrect steps shown: $3x = 11$, $3x3 = 113$, $x = 113$ (These steps are incorrect; always check arithmetic and operations.)

Identities, Contradictions, and Conditional Equations

Types of Linear Equations

• Conditional Equation: Has exactly one solution.

• Contradiction: Has no solution (results in a false statement, e.g., $5 = 3$).

• Identity: Is true for all values of the variable (results in a true statement, e.g., $x = x$).

Example 9: Identifying Equation Types

• $3x + 5 = 3(x + 2)$

• Expand right side: $3x + 5 = 3x + 6$

• Subtract $3x$ from both sides: $5 = 6$

• This is a contradiction (no solution).

• Additional info: If the variable terms cancel and the resulting statement is true (e.g., $5 = 5$), the equation is an identity.

Summary Table: Types of Linear Equations

Type	Description	Example
Conditional	One solution	$2x + 5 = 9$
Contradiction	No solution	$3x + 5 = 3x + 6$
Identity	All real numbers (for which the equation is defined) are solutions	$4x = 4x$