Solving Linear Equations: Study Notes and Practice

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. The solutions to these equations are values of the variable that make the equation true. Mastery of solving linear equations is foundational for all algebraic problem-solving.

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

• Solution: The value of the variable that satisfies the equation.

Steps for Solving Linear Equations

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

• Step 2: Move all variable terms to one side and constants to the other.

• Step 3: Isolate the variable by performing inverse operations.

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

Examples of Solving Linear Equations

• Example 1: $4x + 2 = 3x - 21$ Solution:

  • Subtract $3x$ from both sides: $4x - 3x + 2 = -21$

  • Simplify: $x + 2 = -21$

  • Subtract $2$from both sides:$x = -23$

• Example 2: $7(x + 2) = 6(x - 1)$ Solution:

  • Expand: $7x + 14 = 6x - 6$

  • Subtract $6x$ from both sides: $x + 14 = -6$

  • Subtract $14$from both sides:$x = -20$

Solving Equations with Fractions

When equations contain fractions, it is often helpful to clear the fractions by multiplying both sides by the least common denominator (LCD).

• Step 1: Identify the LCD of all denominators.

• Step 2: Multiply both sides of the equation by the LCD to eliminate fractions.

• Step 3: Solve the resulting linear equation.

Example:

• Equation: $\frac{5x}{3} - \frac{4}{5} = \frac{2x}{15}$ Solution:

  • LCD is 15. Multiply both sides by 15:

  • $15 \left(\frac{5x}{3}\right) - 15 \left(\frac{4}{5}\right) = 15 \left(\frac{2x}{15}\right)$

  • $25x - 12 = 2x$

  • Subtract $2x$ from both sides: $23x - 12 = 0$

  • Add $12$to both sides: $23x = 12$

  • Divide by $23$:$x = \frac{12}{23}$

Practice Problems (from the file)

The following problems are typical examples of linear equations and equations with fractions:

1. $4x + 2 = 3x - 21$

2. $7(x + 2) = 6(x - 1)$

3. $3x - 11 = 4x + 9$

4. $4(x - 7) = 7x - 2$

5. $4(x + 3) = 3(x - 7)$

6. $6(x + 2) = 2x + 7$

7. $0.7x - 2.1 = 3.5x - 2.7$

8. $0.2x - 1.4 = 1.3x - 3.2$

9. $\frac{5x}{3} - \frac{4}{5} = \frac{2x}{15}$

10. $\frac{3x}{4} + \frac{2}{3} = \frac{5x}{6}$

11. $\frac{3x}{5} - \frac{2x}{3} = 4$

12. $\frac{7}{3} = x - \frac{5}{6}$

13. $\frac{2x}{3} - \frac{5}{6} = \frac{x}{2}$

14. $3x - \frac{5}{4} = \frac{4x}{3}$

15. $\frac{2x}{3} + \frac{5}{2} = \frac{x}{2} - 2$

Key Concepts and Formulas

• Isolating the Variable: Use inverse operations to get the variable alone on one side of the equation.

• Combining Like Terms: Add or subtract terms with the same variable.

• Clearing Fractions: Multiply both sides by the LCD.

Summary Table: Steps for Solving Linear Equations

Step	Description
1	Simplify both sides (expand, combine like terms)
2	Move variable terms to one side, constants to the other
3	Isolate the variable
4	Check the solution

Additional info: These notes are based on a worksheet containing practice problems for solving linear equations, including equations with fractions. The content is directly relevant to the "Linear Equations and Inequalities" chapter in Beginning Algebra.