Multi Step Equations With Distributive Property

Mastering Multi-Step Equations: A Comprehensive Guide to the Distributive Property

Solving multi-step equations is a fundamental skill in algebra. These equations require more than one step to isolate the variable and find its solution. One crucial tool in tackling these problems is understanding and applying the distributive property. This comprehensive guide will walk you through the intricacies of multi-step equations, focusing specifically on the role of the distributive property, and providing you with a variety of examples and practice problems.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that states: a(b + c) = ab + ac. This means that you can distribute a term outside parentheses to each term inside the parentheses by multiplying. Similarly, a(b - c) = ab - ac. This seemingly simple property is the key to unlocking many complex multi-step equations.

Example:

3(x + 2) = 15

Using the distributive property, we distribute the 3 to both the x and the 2:

3x + 6 = 15

Now we have a simpler two-step equation that's easy to solve.

Solving Multi-Step Equations with the Distributive Property: A Step-by-Step Approach

Solving multi-step equations involving the distributive property follows a logical sequence of steps:

1. Identify the Distributive Property: Look for parentheses in the equation. If a term is multiplied by a quantity in parentheses, you'll need to apply the distributive property.

2. Distribute: Multiply the term outside the parentheses by each term inside the parentheses. Remember to pay attention to signs – a negative sign outside the parentheses changes the sign of each term inside.

3. Combine Like Terms: After distributing, simplify the equation by combining like terms. This means adding or subtracting terms with the same variable raised to the same power.

4. Isolate the Variable Term: Use inverse operations (addition, subtraction, multiplication, and division) to isolate the term containing the variable on one side of the equation.

5. Solve for the Variable: Finally, isolate the variable by performing the necessary inverse operation to get the variable alone.

6. Check Your Solution: Substitute your solution back into the original equation to verify that it makes the equation true.

Examples: Tackling Different Scenarios

Let's work through some examples to solidify your understanding:

Example 1: Basic Application

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

1. Distribute: 2x + 6 = 10

2. Subtract 6 from both sides: 2x = 4

3. Divide both sides by 2: x = 2

4. Check: 2(2 + 3) = 2(5) = 10. The solution is correct.

Example 2: Negative Distribution

Solve for y: -3(y - 4) = 9

1. Distribute: -3y + 12 = 9

2. Subtract 12 from both sides: -3y = -3

3. Divide both sides by -3: y = 1

4. Check: -3(1 - 4) = -3(-3) = 9. The solution is correct.

Example 3: Combining Like Terms

Solve for z: 4(z + 1) - 2z = 8

1. Distribute: 4z + 4 - 2z = 8

2. Combine like terms: 2z + 4 = 8

3. Subtract 4 from both sides: 2z = 4

4. Divide both sides by 2: z = 2

5. Check: 4(2 + 1) - 2(2) = 12 - 4 = 8. The solution is correct.

Example 4: More Complex Equation

Solve for a: 5(2a - 3) + 4 = 3(a + 2) + 11

1. Distribute: 10a - 15 + 4 = 3a + 6 + 11

2. Combine like terms: 10a - 11 = 3a + 17

3. Subtract 3a from both sides: 7a - 11 = 17

4. Add 11 to both sides: 7a = 28

5. Divide both sides by 7: a = 4

6. Check: 5(2(4) - 3) + 4 = 5(5) + 4 = 29; 3(4 + 2) + 11 = 3(6) + 11 = 29. The solution is correct.

Dealing with Fractions and Decimals

The distributive property works equally well with equations containing fractions and decimals. Remember to follow the order of operations (PEMDAS/BODMAS) and handle fractions and decimals carefully.

Example 5: Equation with Fractions

Solve for b: ½(b + 6) = 4

1. Distribute: ½b + 3 = 4

2. Subtract 3 from both sides: ½b = 1

3. Multiply both sides by 2: b = 2

4. Check: ½(2 + 6) = ½(8) = 4. The solution is correct.

Example 6: Equation with Decimals

Solve for c: 0.2(c + 5) = 3

1. Distribute: 0.2c + 1 = 3

2. Subtract 1 from both sides: 0.2c = 2

3. Divide both sides by 0.2: c = 10

4. Check: 0.2(10 + 5) = 0.2(15) = 3. The solution is correct.

Common Mistakes to Avoid

• Incorrect Distribution: Be meticulous when distributing. Remember to multiply each term inside the parentheses by the term outside. Pay close attention to signs.

• Combining Unlike Terms: Only combine terms with the same variable and exponent.

• Arithmetic Errors: Double-check your calculations throughout the process. A small mistake early on can lead to an incorrect final answer.

• Forgetting to Check Your Solution: Always substitute your solution back into the original equation to confirm its accuracy.

Practice Problems

Here are a few problems for you to practice:

1. 3(x + 5) = 21
2. -2(y - 4) = 6
3. 5(z + 2) - 3z = 11
4. 2(a - 1) + 3a = 11
5. 1/3(b + 9) = 5
6. 0.5(c - 4) = 1

Remember to follow the steps outlined above, distribute carefully, combine like terms, and check your answers. Practice is key to mastering multi-step equations!

Conclusion: Mastering Multi-Step Equations

Multi-step equations are a cornerstone of algebra. Understanding and applying the distributive property is crucial for successfully solving these equations. By following the systematic approach outlined in this guide, paying attention to details, and practicing regularly, you can build a strong foundation in algebra and confidently tackle even more complex equations in the future. Consistent practice and attention to detail will lead to mastery of this important algebraic skill. Good luck!