Unit 1 Equations And Inequalities Homework 3 Solving Equations

Unit 1: Equations and Inequalities – Homework 3: Solving Equations

This comprehensive guide delves into the intricacies of solving equations, a fundamental concept in algebra. We'll cover various techniques, provide numerous examples, and offer strategies to master this crucial skill. Whether you're a high school student tackling your homework or an adult learner brushing up on your algebra skills, this article will serve as your complete resource for conquering equations.

Understanding Equations

Before diving into solving techniques, let's establish a clear understanding of what an equation is. An equation is a mathematical statement asserting the equality of two expressions. These expressions contain variables (usually represented by letters like x, y, or z), constants (numerical values), and mathematical operations (addition, subtraction, multiplication, division, etc.). The goal when solving an equation is to find the value(s) of the variable(s) that make the equation true.

For example, consider the equation 3x + 5 = 14. Here, 'x' is the variable, 3 and 5 are constants, and the '+' symbol represents addition. Solving this equation means finding the value of 'x' that makes the left-hand side (3x + 5) equal to the right-hand side (14).

Basic Techniques for Solving Equations

Solving equations involves manipulating the equation to isolate the variable on one side of the equals sign. This is achieved by applying inverse operations. Remember, whatever operation you perform on one side of the equation, you must perform on the other side to maintain balance.

1. Solving One-Step Equations

These are the simplest types of equations, requiring only one operation to isolate the variable.

Example 1: x + 7 = 12

To isolate 'x', we subtract 7 from both sides:

x + 7 - 7 = 12 - 7

x = 5

Example 2: y - 3 = 9

To isolate 'y', we add 3 to both sides:

y - 3 + 3 = 9 + 3

y = 12

Example 3: 4z = 20

To isolate 'z', we divide both sides by 4:

4z / 4 = 20 / 4

z = 5

Example 4: w/5 = 2

To isolate 'w', we multiply both sides by 5:

(w/5) * 5 = 2 * 5

w = 10

2. Solving Two-Step Equations

These equations require two operations to isolate the variable. The order of operations generally involves dealing with addition/subtraction first, then multiplication/division.

Example 5: 2x + 3 = 7

1. Subtract 3 from both sides: 2x + 3 - 3 = 7 - 3 => 2x = 4
2. Divide both sides by 2: 2x / 2 = 4 / 2 => x = 2

Example 6: 5y - 8 = 17

1. Add 8 to both sides: 5y - 8 + 8 = 17 + 8 => 5y = 25
2. Divide both sides by 5: 5y / 5 = 25 / 5 => y = 5

Example 7: (z/3) + 2 = 6

1. Subtract 2 from both sides: (z/3) + 2 - 2 = 6 - 2 => z/3 = 4
2. Multiply both sides by 3: (z/3) * 3 = 4 * 3 => z = 12

Example 8: 4w - 5 = 11

1. Add 5 to both sides: 4w -5 + 5 = 11 + 5 => 4w = 16
2. Divide both sides by 4: 4w/4 = 16/4 => w = 4

3. Solving Equations with Variables on Both Sides

These equations have variables on both the left and right sides of the equals sign. The first step is to combine like terms by moving all variable terms to one side and constant terms to the other.

Example 9: 3x + 5 = 2x + 10

1. Subtract 2x from both sides: 3x - 2x + 5 = 2x - 2x + 10 => x + 5 = 10
2. Subtract 5 from both sides: x + 5 - 5 = 10 - 5 => x = 5

Example 10: 7y - 2 = 4y + 10

1. Subtract 4y from both sides: 7y - 4y - 2 = 4y - 4y + 10 => 3y - 2 = 10
2. Add 2 to both sides: 3y - 2 + 2 = 10 + 2 => 3y = 12
3. Divide both sides by 3: 3y / 3 = 12 / 3 => y = 4

Example 11: 5z + 8 = 2z – 1

1. Subtract 2z from both sides: 5z - 2z + 8 = 2z - 2z - 1 => 3z + 8 = -1
2. Subtract 8 from both sides: 3z + 8 - 8 = -1 - 8 => 3z = -9
3. Divide both sides by 3: 3z/3 = -9/3 => z = -3

4. Solving Equations with Parentheses

Equations with parentheses require distributing any terms outside the parentheses before proceeding with the usual solving steps.

Example 12: 2(x + 3) = 10

1. Distribute the 2: 2 * x + 2 * 3 = 10 => 2x + 6 = 10
2. Subtract 6 from both sides: 2x + 6 - 6 = 10 - 6 => 2x = 4
3. Divide both sides by 2: 2x / 2 = 4 / 2 => x = 2

Example 13: 3(y - 2) + 4 = 13

1. Distribute the 3: 3 * y - 3 * 2 + 4 = 13 => 3y - 6 + 4 = 13
2. Simplify: 3y - 2 = 13
3. Add 2 to both sides: 3y - 2 + 2 = 13 + 2 => 3y = 15
4. Divide both sides by 3: 3y / 3 = 15 / 3 => y = 5

Example 14: 5(2z + 1) - 3z = 22

1. Distribute the 5: 10z + 5 - 3z = 22
2. Combine like terms: 7z + 5 = 22
3. Subtract 5 from both sides: 7z + 5 - 5 = 22 - 5 => 7z = 17
4. Divide both sides by 7: 7z/7 = 17/7 => z = 17/7

Solving Inequalities

Inequalities are similar to equations, but instead of an equals sign (=), they use inequality symbols:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

Solving inequalities involves the same principles as solving equations, with one crucial exception: when multiplying or dividing both sides by a negative number, you must reverse the inequality symbol.

Example 15: x + 4 > 10

Subtract 4 from both sides: x > 6

Example 16: 2y - 3 ≤ 7

1. Add 3 to both sides: 2y ≤ 10
2. Divide both sides by 2: y ≤ 5

Example 17: -3z + 6 < 9

1. Subtract 6 from both sides: -3z < 3
2. Divide both sides by -3 and reverse the inequality symbol: z > -1

Example 18: 4w - 7 ≥ 13

1. Add 7 to both sides: 4w ≥ 20
2. Divide both sides by 4: w ≥ 5

Advanced Techniques and Problem-Solving Strategies

While the above techniques cover a majority of equation-solving scenarios, some problems require more advanced approaches. Here are a few strategies:

1. Clearing Fractions and Decimals

Equations containing fractions or decimals can be simplified by multiplying both sides by the least common denominator (LCD) or a power of 10, respectively.

2. Using the Distributive Property Effectively

Careful and accurate application of the distributive property is vital, especially when dealing with complex expressions. Remember to distribute to every term within the parentheses.

3. Checking Your Solutions

Always check your solution by substituting it back into the original equation to ensure it satisfies the equality (or inequality). This step helps identify errors in your calculations.

4. Identifying and Avoiding Common Mistakes

• Incorrect order of operations: Always follow PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Forgetting to distribute: Ensure you distribute correctly to all terms within parentheses.
• Incorrectly handling negative signs: Pay close attention to signs when adding, subtracting, multiplying, or dividing negative numbers.
• Not reversing the inequality sign: Remember to reverse the inequality symbol when multiplying or dividing by a negative number.

Conclusion

Mastering the art of solving equations and inequalities is a cornerstone of algebraic proficiency. By understanding the fundamental techniques, practicing regularly with diverse examples, and employing effective problem-solving strategies, you can build a strong foundation in algebra and confidently tackle more complex mathematical concepts. Remember to practice consistently, and don't hesitate to seek help when needed. With dedication and the right approach, you'll become proficient in solving equations and inequalities. Good luck with your homework!