Unit 1 Equations & Inequalities Homework 3 Solving Equations

Solving equations is a fundamental skill in algebra and beyond. It's the process of finding the value(s) of a variable that make the equation true. This homework assignment focuses on mastering various techniques for solving different types of equations, encompassing linear equations, equations with variables on both sides, and those involving the distributive property.

Understanding the Basics

Before diving into complex problems, it's crucial to grasp the core principles of solving equations. The primary goal is to isolate the variable on one side of the equation. We achieve this by performing inverse operations on both sides, maintaining the equality. Here’s a review of key concepts:

• Equation: A mathematical statement that asserts the equality of two expressions. It contains an equals sign (=).
• Variable: A symbol (usually a letter like x, y, or z) representing an unknown value.
• Constant: A fixed numerical value in an equation.
• Coefficient: A number multiplying a variable (e.g., in 3x, the coefficient is 3).
• Inverse Operations: Operations that undo each other (e.g., addition and subtraction, multiplication and division).

Key Properties of Equality

To manipulate equations correctly, we rely on these properties:

• Addition Property of Equality: If a = b, then a + c = b + c. (Adding the same value to both sides doesn't change the equality.)
• Subtraction Property of Equality: If a = b, then a - c = b - c. (Subtracting the same value from both sides doesn't change the equality.)
• Multiplication Property of Equality: If a = b, then ac = bc. (Multiplying both sides by the same non-zero value doesn't change the equality.)
• Division Property of Equality: If a = b, then a/c = b/c, provided c ≠ 0. (Dividing both sides by the same non-zero value doesn't change the equality.)
• Distributive Property: a(b + c) = ab + ac. (Allows us to multiply a term across a sum or difference.)

Solving Linear Equations: Step-by-Step

Linear equations are equations where the highest power of the variable is 1. They can be solved using the following general steps:

1. Simplify both sides: Combine like terms and use the distributive property to eliminate parentheses.
2. Isolate the variable term: Use addition or subtraction to get all terms containing the variable on one side of the equation and all constant terms on the other side.
3. Isolate the variable: Use multiplication or division to solve for the variable.
4. Check your solution: Substitute the value you found for the variable back into the original equation to ensure it's a valid solution.

Example 1: Solve for x: 3x + 5 = 14

• Step 1: The equation is already simplified.
• Step 2: Subtract 5 from both sides: 3x + 5 - 5 = 14 - 5 => 3x = 9
• Step 3: Divide both sides by 3: 3x / 3 = 9 / 3 => x = 3
• Step 4: Check: 3(3) + 5 = 9 + 5 = 14. The solution is correct.

Example 2: Solve for y: 2y - 7 = -1

• Step 1: The equation is already simplified.
• Step 2: Add 7 to both sides: 2y - 7 + 7 = -1 + 7 => 2y = 6
• Step 3: Divide both sides by 2: 2y / 2 = 6 / 2 => y = 3
• Step 4: Check: 2(3) - 7 = 6 - 7 = -1. The solution is correct.

Solving Equations with Variables on Both Sides

When the variable appears on both sides of the equation, you need to strategically collect the variable terms on one side before isolating the variable itself.

1. Simplify both sides: As before, combine like terms and use the distributive property.
2. Move variable terms to one side: Add or subtract the variable term from one side to eliminate it from the other. Generally, it's easier to move the smaller variable term to the side with the larger variable term to avoid negative coefficients.
3. Move constant terms to the other side: Add or subtract constant terms to isolate them on the side opposite the variable terms.
4. Isolate the variable: Use multiplication or division to solve for the variable.
5. Check your solution: Substitute the value you found back into the original equation.

Example 3: Solve for a: 5a + 3 = 2a - 6

• Step 1: The equation is already simplified.
• Step 2: Subtract 2a from both sides: 5a + 3 - 2a = 2a - 6 - 2a => 3a + 3 = -6
• Step 3: Subtract 3 from both sides: 3a + 3 - 3 = -6 - 3 => 3a = -9
• Step 4: Divide both sides by 3: 3a / 3 = -9 / 3 => a = -3
• Step 5: Check: 5(-3) + 3 = -15 + 3 = -12 and 2(-3) - 6 = -6 - 6 = -12. The solution is correct.

Example 4: Solve for z: 7z - 4 = -3z + 16

• Step 1: The equation is already simplified.
• Step 2: Add 3z to both sides: 7z - 4 + 3z = -3z + 16 + 3z => 10z - 4 = 16
• Step 3: Add 4 to both sides: 10z - 4 + 4 = 16 + 4 => 10z = 20
• Step 4: Divide both sides by 10: 10z / 10 = 20 / 10 => z = 2
• Step 5: Check: 7(2) - 4 = 14 - 4 = 10 and -3(2) + 16 = -6 + 16 = 10. The solution is correct.

Solving Equations Using the Distributive Property

The distributive property is essential when dealing with equations that contain parentheses. Remember, a(b + c) = ab + ac.

1. Distribute: Apply the distributive property to eliminate parentheses. Multiply the term outside the parentheses by each term inside the parentheses.
2. Simplify both sides: Combine like terms after distributing.
3. Isolate the variable term: Move all terms containing the variable to one side of the equation.
4. Isolate the variable: Divide both sides by the coefficient of the variable.
5. Check your solution: Substitute your answer back into the original equation.

Example 5: Solve for x: 2(x + 3) = 10

• Step 1: Distribute the 2: 2 * x + 2 * 3 = 10 => 2x + 6 = 10
• Step 2: The equation is simplified after distributing.
• Step 3: Subtract 6 from both sides: 2x + 6 - 6 = 10 - 6 => 2x = 4
• Step 4: Divide both sides by 2: 2x / 2 = 4 / 2 => x = 2
• Step 5: Check: 2(2 + 3) = 2(5) = 10. The solution is correct.

Example 6: Solve for m: -3(m - 2) = 18

• Step 1: Distribute the -3: -3 * m + (-3) * (-2) = 18 => -3m + 6 = 18
• Step 2: The equation is simplified after distributing.
• Step 3: Subtract 6 from both sides: -3m + 6 - 6 = 18 - 6 => -3m = 12
• Step 4: Divide both sides by -3: -3m / -3 = 12 / -3 => m = -4
• Step 5: Check: -3(-4 - 2) = -3(-6) = 18. The solution is correct.

Example 7: Solve for p: 4(2p - 1) = 3(p + 2)

• Step 1: Distribute on both sides: 4 * 2p + 4 * (-1) = 3 * p + 3 * 2 => 8p - 4 = 3p + 6
• Step 2: The equation is simplified after distributing.
• Step 3: Subtract 3p from both sides: 8p - 4 - 3p = 3p + 6 - 3p => 5p - 4 = 6
• Step 4: Add 4 to both sides: 5p - 4 + 4 = 6 + 4 => 5p = 10
• Step 5: Divide both sides by 5: 5p / 5 = 10 / 5 => p = 2
• Step 6: Check: 4(2(2) - 1) = 4(4 - 1) = 4(3) = 12 and 3(2 + 2) = 3(4) = 12. The solution is correct.

Dealing with Fractions and Decimals

Equations involving fractions or decimals may seem intimidating, but they can be solved by eliminating the fractions or decimals first.

Eliminating Fractions

1. Find the Least Common Denominator (LCD): Determine the LCD of all the fractions in the equation.
2. Multiply all terms by the LCD: Multiply every term on both sides of the equation by the LCD. This will cancel out the denominators.
3. Simplify: Simplify the equation by canceling the denominators and performing any necessary multiplications.
4. Solve the equation: Solve the resulting equation using the techniques described above.
5. Check your solution: Substitute your answer back into the original equation.

Example 8: Solve for x: (x/2) + (1/3) = (5/6)

• Step 1: The LCD of 2, 3, and 6 is 6.
• Step 2: Multiply all terms by 6: 6 * (x/2) + 6 * (1/3) = 6 * (5/6)
• Step 3: Simplify: 3x + 2 = 5
• Step 4: Solve for x:
  • Subtract 2 from both sides: 3x = 3
  • Divide both sides by 3: x = 1
• Step 5: Check: (1/2) + (1/3) = (3/6) + (2/6) = (5/6). The solution is correct.

Eliminating Decimals

1. Identify the decimal with the most decimal places: Determine the maximum number of decimal places present in any term of the equation.
2. Multiply all terms by a power of 10: Multiply every term on both sides of the equation by 10 raised to the power of the maximum number of decimal places. This will shift the decimal point to the right, eliminating the decimals.
3. Simplify: Simplify the equation by performing the multiplications.
4. Solve the equation: Solve the resulting equation using the techniques described above.
5. Check your solution: Substitute your answer back into the original equation.

Example 9: Solve for y: 0.2y + 1.5 = 2.1

• Step 1: The maximum number of decimal places is 1.
• Step 2: Multiply all terms by 10: 10 * (0.2y) + 10 * (1.5) = 10 * (2.1)
• Step 3: Simplify: 2y + 15 = 21
• Step 4: Solve for y:
  • Subtract 15 from both sides: 2y = 6
  • Divide both sides by 2: y = 3
• Step 5: Check: 0.2(3) + 1.5 = 0.6 + 1.5 = 2.1. The solution is correct.

Special Cases: Identities and Contradictions

Not all equations have a unique solution. Some equations are identities, while others are contradictions.

• Identity: An equation that is true for all values of the variable. When you solve an identity, the variable will disappear, and you'll be left with a true statement (e.g., 0 = 0, 5 = 5). This indicates that any value of the variable will satisfy the equation.

Example 10: Solve for x: 2(x + 3) = 2x + 6

• Step 1: Distribute: 2x + 6 = 2x + 6
• Step 2: Subtract 2x from both sides: 6 = 6

Since 6 = 6 is always true, this equation is an identity. The solution is all real numbers.

• Contradiction: An equation that is never true for any value of the variable. When you solve a contradiction, the variable will disappear, and you'll be left with a false statement (e.g., 0 = 5, -2 = 3). This indicates that there is no solution to the equation.

Example 11: Solve for y: 3(y - 1) = 3y + 2

• Step 1: Distribute: 3y - 3 = 3y + 2
• Step 2: Subtract 3y from both sides: -3 = 2

Since -3 = 2 is never true, this equation is a contradiction. There is no solution.

Strategies for Success

Solving equations effectively requires practice and a strategic approach. Here are some tips:

• Show your work: Write down each step clearly. This makes it easier to track your progress, identify errors, and learn from your mistakes.
• Be organized: Keep your work neat and organized. This helps prevent errors and makes it easier to review your work.
• Check your answers: Always check your solutions by substituting them back into the original equation. This is the best way to ensure that you have found the correct solution.
• Practice regularly: The more you practice, the more comfortable and confident you will become in solving equations.
• Don't be afraid to ask for help: If you are struggling with a particular type of equation, don't hesitate to ask your teacher, a tutor, or a classmate for help.
• Understand the "why" not just the "how": Focus on understanding the underlying principles of equation solving rather than just memorizing steps. This will help you solve more complex problems and apply your knowledge in different contexts.
• Break down complex problems: If an equation seems overwhelming, break it down into smaller, more manageable steps.
• Use different methods: Sometimes, there are multiple ways to solve an equation. Experiment with different methods to find the one that works best for you.
• Pay attention to detail: Even a small error can lead to an incorrect solution. Be careful when performing operations and combining like terms.

Common Mistakes to Avoid

Many students make common mistakes when solving equations. Being aware of these mistakes can help you avoid them.

• Incorrectly applying the distributive property: Make sure to multiply the term outside the parentheses by every term inside the parentheses. Also, pay attention to signs.
• Combining unlike terms: Only combine terms that have the same variable and exponent.
• Performing operations on only one side of the equation: Remember that to maintain equality, you must perform the same operation on both sides of the equation.
• Forgetting to check your answers: Always check your solutions to ensure they are correct.
• Dividing by zero: Division by zero is undefined. Be careful not to divide by zero when solving equations.
• Incorrectly handling negative signs: Pay close attention to negative signs, especially when distributing or combining like terms.

Real-World Applications

Solving equations is not just an abstract mathematical skill. It has many real-world applications in various fields, including:

• Physics: Calculating motion, forces, and energy.
• Engineering: Designing structures, circuits, and machines.
• Chemistry: Balancing chemical equations and calculating reaction rates.
• Economics: Modeling supply and demand, and analyzing financial data.
• Computer Science: Developing algorithms and solving problems in programming.
• Everyday life: Budgeting, calculating discounts, and determining distances.

Conclusion

Mastering the art of solving equations is a cornerstone of mathematical proficiency. By understanding the underlying principles, practicing consistently, and avoiding common mistakes, you can develop the skills necessary to tackle a wide range of algebraic problems. This homework assignment provides a valuable opportunity to hone your equation-solving abilities and build a solid foundation for future mathematical endeavors. Remember to focus on understanding the "why" behind each step, not just the "how," and you'll be well on your way to success. Embrace the challenge, persevere through difficulties, and enjoy the satisfaction of finding the solution.