Solving Linear Equations

Introduction to Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. Solving linear equations is a foundational skill in algebra, requiring the application of several systematic steps to isolate the variable and determine its value.

• Linear Equation: An equation of the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

• Solution: The value of the variable that makes the equation true.

Steps for Solving Linear Equations

To solve a linear equation, follow these steps:

1. Distribute constants: Apply the distributive property to remove parentheses. $a(b + c) = ab + ac$

2. Combine like terms: Add or subtract terms with the same variable or constants on each side of the equation.

3. Group terms: Move all terms containing the variable to one side and constants to the other side.

4. Isolate the variable: Solve for the variable by performing inverse operations.

5. Check the solution: Substitute the solution back into the original equation to verify correctness.

Example:

Solve the equation $2(x - 3) = 0$

• Distribute: $2x - 6 = 0$

• Add 6 to both sides: $2x = 6$

• Divide by 2: $x = 3$

• Check: $2(3 - 3) = 2(0) = 0$ (True)

Solving Linear Equations with Fractions

When linear equations contain fractions, it is often helpful to eliminate the fractions first by multiplying both sides by the Least Common Denominator (LCD).

1. Multiply by LCD: Multiply both sides of the equation by the LCD to clear fractions.

2. Continue with the standard steps: Distribute, combine like terms, group terms, and isolate the variable.

Example:

Solve the equation $\frac{1}{4}(x + 2) - \frac{1}{3}x = \frac{1}{12}$

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

• $12 \times \left[ \frac{1}{4}(x + 2) - \frac{1}{3}x \right] = 12 \times \frac{1}{12}$

• $3(x + 2) - 4x = 1$

• $3x + 6 - 4x = 1$

• $-x + 6 = 1$

• $-x = 1 - 6 = -5$

• $x = 5$

• Check: $\frac{1}{4}(5 + 2) - \frac{1}{3}(5) = \frac{1}{4}(7) - \frac{5}{3} = \frac{7}{4} - \frac{5}{3}$ Find common denominator (12): $\frac{21}{12} - \frac{20}{12} = \frac{1}{12}$ (True)

Practice Problem

Solve the equation $3(2 - 5x) = 4x + 25$

• Distribute: $6 - 15x = 4x + 25$

• Group $x$ terms: $6 - 15x - 4x = 25$

• $6 - 19x = 25$

• Subtract 6: $-19x = 19$

• Divide by -19: $x = -1$

• Check: $3(2 - 5(-1)) = 4(-1) + 25$ $3(2 + 5) = -4 + 25$ $3(7) = 21$ and $-4 + 25 = 21$ (True)

Categorizing Linear Equations by Number of Solutions

Linear equations can be classified based on the number of solutions they possess:

When solving, always check if the equation simplifies to a contradiction (no solution) or an identity (all real numbers are solutions).

Summary of Key Steps

• Distribute constants to remove parentheses.

• Combine like terms on each side.

• Group variable terms on one side and constants on the other.

• Isolate the variable using inverse operations.

• Check your solution by substitution.

• For equations with fractions, multiply both sides by the LCD first.

• Classify the equation based on the number of solutions.

Additional info: The notes infer standard steps and terminology for solving linear equations, including the classification of equations by solution type, based on common college algebra curricula.