Solve For W 5w 9z 2z 3w

Solving for 'w': A Comprehensive Guide to Algebraic Manipulation

This article provides a comprehensive guide on how to solve for the variable 'w' in the equation 5w + 9z + 2z = 3w. We'll break down the process step-by-step, explaining the underlying algebraic principles involved. This will not only help you solve this specific equation but will also equip you with the skills to tackle similar algebraic problems. We'll also explore variations and extensions of this problem to solidify your understanding.

Understanding the Equation: 5w + 9z + 2z = 3w

Before we begin solving, let's analyze the given equation: 5w + 9z + 2z = 3w. This is a linear equation with two variables, 'w' and 'z'. Our goal is to isolate 'w' on one side of the equation, expressing it in terms of 'z'. This involves using algebraic manipulation techniques to move terms around while maintaining the equality of the equation.

Step-by-Step Solution: Isolating 'w'

Here's a detailed breakdown of the steps involved in solving for 'w':

Step 1: Combine Like Terms

The equation contains like terms involving 'z'. We can simplify the equation by combining these terms:

9z + 2z = 11z

This simplifies our equation to:

5w + 11z = 3w

Step 2: Move 'w' Terms to One Side

To isolate 'w', we need to move all terms containing 'w' to one side of the equation and all other terms to the other side. We can subtract 3w from both sides:

5w - 3w + 11z = 3w - 3w

This simplifies to:

2w + 11z = 0

Step 3: Isolate 'w'

Now, we need to isolate 'w'. We can achieve this by subtracting 11z from both sides:

2w + 11z - 11z = 0 - 11z

This gives us:

2w = -11z

Step 4: Solve for 'w'

Finally, we divide both sides by 2 to solve for 'w':

(2w)/2 = (-11z)/2

This results in our solution:

w = -11z/2 or w = -5.5z

This means the value of 'w' is directly dependent on the value of 'z'. For any given value of 'z', we can calculate the corresponding value of 'w' using this equation.

Verification: Checking Your Solution

It's crucial to verify your solution. Let's substitute our solution back into the original equation:

5w + 9z + 2z = 3w

Substitute w = -11z/2:

5(-11z/2) + 9z + 2z = 3(-11z/2)

Simplify:

-55z/2 + 11z = -33z/2

Find a common denominator:

(-55z + 22z)/2 = -33z/2

-33z/2 = -33z/2

The equation holds true, confirming our solution is correct.

Expanding Your Understanding: Variations and Extensions

Let's explore variations of this problem to strengthen your understanding of algebraic manipulation:

Variation 1: Different Coefficients

Consider the equation: 7w + 5z - 3z = 2w + 10. This problem introduces different coefficients for 'w' and 'z' and also includes a constant term. The steps to solve would be similar:

1. Combine like terms: 7w + 2z = 2w + 10
2. Move 'w' terms to one side: 5w + 2z = 10
3. Move 'z' term to the other side: 5w = 10 - 2z
4. Solve for 'w': w = (10 - 2z)/5

Variation 2: Equations with More Variables

Imagine an equation with three variables: 2w + 3x + 4z = 10. Solving for 'w' in this scenario requires a slightly different approach. You would need to express 'w' in terms of 'x' and 'z'. The solution would be:

w = (10 - 3x - 4z)/2

Variation 3: Equations with Exponents

Equations involving exponents introduce a new level of complexity. Consider: w² + 2z = 5. Solving for 'w' would require taking the square root:

w² = 5 - 2z w = ±√(5 - 2z)

Note the ± symbol, indicating that there are two possible solutions for 'w' in this case.

Practical Applications: Why is Solving for Variables Important?

The ability to solve for variables is fundamental in many fields, including:

• Physics: Solving equations of motion, calculating forces, and understanding energy transfer.
• Engineering: Designing structures, analyzing circuits, and modeling systems.
• Economics: Creating models of supply and demand, analyzing market trends, and forecasting economic growth.
• Computer Science: Developing algorithms, optimizing code, and solving computational problems.
• Data Science and Statistics: Analyzing data, building predictive models, and drawing inferences.

Solving for variables is a crucial skill that underpins many complex calculations and problem-solving scenarios.

Conclusion: Mastering Algebraic Manipulation

Solving for 'w' in the equation 5w + 9z + 2z = 3w, and understanding the underlying principles, empowers you to tackle more complex algebraic problems. Remember that the key steps are combining like terms, moving variables to one side, and isolating the variable you're solving for. Practice different variations and extensions to solidify your understanding and build confidence in your ability to manipulate algebraic expressions. The more you practice, the more proficient you'll become in this fundamental skill. This skill is vital in various academic disciplines and professional fields, making it a valuable asset to develop.