Solving Linear Equations: Study Notes and Practice Problems

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 general form of a linear equation in one variable is ax + b = c, where a, b, and c are constants.

• Linear equations have at most one solution for the variable.

• Solving a linear equation involves isolating the variable on one side of the equation.

• Equations may include parentheses, fractions, or decimals, which require additional steps to simplify.

Steps for Solving Linear Equations

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

2. Isolate the variable by using addition, subtraction, multiplication, or division.

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

Examples and Applications

• Example 1: $7x - (3x - 1) = 2$ Step 1: Distribute and combine like terms: $7x - 3x + 1 = 2$ Step 2: $4x + 1 = 2$ Step 3: $4x = 1$ Step 4: $x = \frac{1}{4}$

• Example 2: $\frac{5x}{2} - 4 = 1$ Step 1: Add 4 to both sides: $\frac{5x}{2} = 5$ Step 2: Multiply both sides by 2: $5x = 10$ Step 3: $x = 2$

Solving Equations with Fractions and Decimals

When equations contain fractions or decimals, it is often helpful to clear them by multiplying both sides by the least common denominator (LCD) or by an appropriate power of 10.

• Example: $\frac{x}{2} + \frac{x}{3} = 5$ Step 1: LCD is 6. Multiply both sides by 6: $3x + 2x = 30$ Step 2: $5x = 30$ Step 3: $x = 6$

• Example: $1.3x + 0.9x = 1.5$ Step 1: Combine like terms: $2.2x = 1.5$ Step 2: $x = \frac{1.5}{2.2}$

Practice Problems

The following problems provide practice in solving linear equations, including those with fractions and decimals:

1. $7x - (3x - 1) = 2$

2. $9(4x - 1) = 24$

3. $(y - 2) - (y + 8) = 4y$

4. $3(5x + 2) = 9$

5. $2(4x + 3y) = 52$

6. $-2(2x - 6) = 4$

7. $1.3x + 0.9x = 1.5$

8. $1.4x - 1.1x = 0.2x + 1.3$

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

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

11. $\frac{x}{7} = 10$

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

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

14. $\frac{3x}{2} = \frac{5x}{3}$

15. $\frac{2x}{3} = \frac{3x}{2}$

Key Terms

• Linear Equation: An equation of the form $ax + b = c$.

• Variable: A symbol, usually a letter, that represents an unknown value.

• Coefficient: The numerical factor multiplied by the variable.

• Constant: A fixed value that does not change.

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

Summary Table: Common Steps in Solving Linear Equations

Step	Description
Simplify	Remove parentheses, combine like terms
Isolate Variable	Use addition/subtraction to move terms
Clear Fractions/Decimals	Multiply both sides by LCD or power of 10
Solve	Divide or multiply to find the variable
Check	Substitute solution back into original equation

Additional info: These notes expand on the practice worksheet by providing definitions, examples, and a summary table for exam preparation.