Solving Linear Equations in One Variable

Solving Linear Equations in One Variable

Introduction to Linear Equations

Solving a linear equation involves finding the value of the variable that makes the equation a true statement. This value is called the solution or root of the equation.

Properties Used in Solving Equations

• Addition Property of Equality: If two expressions are equal, adding the same value to both sides preserves equality.

  • If $a = b$, then $a + c = b + c$

• Multiplication Property of Equality: If two expressions are equal, multiplying both sides by the same nonzero value preserves equality.

  • If $a = b$, then $a \cdot c = b \cdot c$

Solving Linear Equations: Examples

• Example 1: Solve $5(x - 1) = 8 - 2x$

  • Distribute and collect like terms to isolate $x$.

  • Apply the properties of equality as needed.

• Example 2: Solve $x - \frac{2}{3} = 2 - \frac{1}{5}x$

  • Clear fractions by multiplying both sides by the least common denominator.

  • Combine like terms and solve for $x$.

• Example 3: Solve $-3(5s + 4) + 8 = 2(2s - 3) + 10$

  • Expand both sides and collect like terms.

  • Isolate $s$ to find the solution.

• Example 4: Solve $-4.0x + 2.8 - 1.6x = 1.3 - 5.7x + 1.4$

  • Combine like terms and solve for $x$.

• Example 5: Solve $7y - 1 = -1 + 7y$

  • Move all terms involving $y$ to one side and constants to the other.

  • Check for special cases (see below).

• Example 6: Solve $x - \frac{-5}{8} = 4 - \frac{2}{3}x$

  • Clear fractions and solve for $x$.

Types of Solutions for Linear Equations

When solving a linear equation, there are three possible outcomes:

1. Conditional Equation: The equation has one solution (one value that makes it true). Example: $2x + 3 = 7$ has the solution $x = 2$.

2. Contradiction (Inconsistent Equation): The equation has no solution (no value makes it true). Example: $x + 2 = x + 5$ leads to $2 = 5$, which is never true.

3. Identity: The equation is true for all real numbers (any value of the variable makes it true). Example: $2(x + 1) = 2x + 2$ simplifies to $2x + 2 = 2x + 2$.

Summary Table: Types of Linear Equations

Type	Description	Number of Solutions	Example
Conditional	True for one value of the variable	One	$x + 3 = 5$
Contradiction (Inconsistent)	Never true	None	$x + 2 = x + 5$
Identity	True for all real numbers	Infinite	$2(x + 1) = 2x + 2$

Key Steps in Solving Linear Equations

1. Simplify both sides of the equation (expand parentheses, combine like terms).

2. Use the Addition or Subtraction Property of Equality to collect variable terms on one side and constants on the other.

3. Use the Multiplication or Division Property of Equality to solve for the variable.

4. Check your solution by substituting it back into the original equation.

Additional info:

• When equations contain fractions, multiply both sides by the least common denominator (LCD) to clear fractions before solving.

• Always check for special cases where variables cancel out, which may indicate an identity or contradiction.