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. These equations are fundamental in algebra and are used to find the value of unknown variables.

• Standard Form: A linear equation in one variable can be written as ax + b = c, where a, b, and c are constants.

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

Steps for Solving Linear Equations

1. Simplify both sides of the equation by removing parentheses and combining like terms.

2. Isolate the variable on one side of the equation using addition or subtraction.

3. Solve for the variable by dividing or multiplying both sides by the coefficient of the variable.

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

Examples

• Example 1: Solve $4(x + 2) = 3(x - 2)$

  • Expand: $4x + 8 = 3x - 6$

  • Subtract $3x$ from both sides: $x + 8 = -6$

  • Subtract $8$from both sides:$x = -14$

• Example 2: Solve $3x - 2 = 5x + 4$

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

  • Subtract $4$from both sides:$-6 = 2x$

  • Divide both sides by $2$:$x = -3$

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 in the equation.

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

• Step 3: Solve the resulting linear equation as usual.

• Example: Solve $\frac{x}{3} + \frac{4}{5} = 2$

  • LCD is 15. Multiply both sides by 15:

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

  • $5x + 12 = 30$

  • $5x = 18$

  • $x = \frac{18}{5}$

Key Properties of Linear Equations

• Addition Property: If $a = b$, then $a + c = b + c$ for any $c$.

• Subtraction Property: If $a = b$, then $a - c = b - c$ for any $c$.

• Multiplication Property: If $a = b$, then $a \times c = b \times c$ for any $c$.

• Division Property: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

Practice Problems (from the file)

The following are representative problems for practice, covering equations with variables on both sides, equations with parentheses, and equations with fractions:

1. $9(x + 2) = 3(x - 2)$

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

3. $3x - 11 = 9x + 7$

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

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

6. $6(x + 3) = 5(x + 7)$

7. $0.7x - 2.1 = 5x + 2.7$

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

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

10. $\frac{3x}{4} - \frac{2}{5} = 1$

11. $\frac{1}{2}x + \frac{1}{3}x = 5$

12. $\frac{7}{5} - x = \frac{3}{10}$

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

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

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

Summary Table: Steps for Solving Linear Equations

Step	Description	Example
1	Simplify both sides	$2(x + 3) = 2x + 6$
2	Move variables to one side	$2x + 6 = 2x + 6$
3	Isolate the variable	$0 = 0$ (infinite solutions) or $x = a$
4	Check the solution	Substitute $x$ back into the original equation

Additional info: These notes are based on a worksheet of practice problems for solving linear equations, including equations with fractions and decimals. The content is foundational for students in Beginning Algebra.