Solve For W 5w 9z 2z 3w

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

This article provides a detailed explanation of how to solve for the variable 'w' in the equation 5w + 9z + 2z = 3w. We'll break down the process step-by-step, covering the fundamental algebraic principles involved and offering additional tips for solving similar equations. Understanding this process is crucial for mastering fundamental algebra and tackling more complex problems.

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

Before diving into the solution, 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 means we want to manipulate the equation until we have 'w = [some expression involving z]'.

Step-by-Step Solution: Isolating 'w'

Here's how we solve for 'w' in a methodical and easy-to-understand manner:

Step 1: Combine Like Terms

The first step involves simplifying the equation by combining like terms. Notice that we have two terms involving 'z': 9z and 2z. Combining these gives us 11z. The equation now becomes:

5w + 11z = 3w

Step 2: Move 'w' Terms to One Side

To isolate 'w', we need to get all terms containing 'w' on one side of the equation and all other terms on the opposite side. Let's 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'. To do this, subtract 11z from both sides:

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

This simplifies to:

2w = -11z

Step 4: Solve for 'w'

Finally, to get 'w' by itself, divide both sides of the equation by 2:

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

This gives us the solution:

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

Therefore, the solution for 'w' in the equation 5w + 9z + 2z = 3w is w = -5.5z. This means the value of 'w' depends directly on the value of 'z'. If you know the value of 'z', you can easily calculate the corresponding value of 'w' using this equation.

Verifying the Solution

It's always a good practice to verify your solution. Let's substitute our solution, w = -5.5z, back into the original equation:

5w + 9z + 2z = 3w

5(-5.5z) + 9z + 2z = 3(-5.5z)

-27.5z + 11z = -16.5z

-16.5z = -16.5z

The equation holds true, confirming that our solution, w = -5.5z, is correct.

Expanding on Algebraic Concepts

This simple equation exemplifies several key algebraic concepts that are fundamental to more advanced mathematical problems. Let's delve into some of these concepts:

Linear Equations

The equation 5w + 9z + 2z = 3w is a linear equation. This means that the highest power of any variable is 1. Linear equations represent straight lines when graphed. Solving linear equations involves manipulating the equation to isolate the variable of interest.

Variables and Constants

In the equation, 'w' and 'z' are variables, representing unknown quantities. The numbers 5, 9, 2, and 3 are constants, representing fixed values.

Like Terms

Combining like terms is a crucial step in simplifying equations. Like terms are terms that have the same variables raised to the same powers. For example, 9z and 2z are like terms because they both have the variable 'z' raised to the power of 1.

Inverse Operations

Solving for 'w' involved using inverse operations. Addition and subtraction are inverse operations, as are multiplication and division. We used subtraction to move terms and division to isolate 'w'.

Properties of Equality

Throughout the solution process, we applied the properties of equality. These properties state that if you perform the same operation on both sides of an equation, the equation remains balanced. For example, subtracting 3w from both sides or dividing both sides by 2 maintains the equality.

Solving Similar Equations: Practice Problems

To solidify your understanding, let's look at some similar equations and how to solve them:

Problem 1: 7x + 4y - 2y = 2x

Solution:

1. Combine like terms: 7x + 2y = 2x
2. Subtract 2x from both sides: 5x + 2y = 0
3. Subtract 2y from both sides: 5x = -2y
4. Divide both sides by 5: x = -2y/5

Problem 2: 3a + 6b - b = 10a

Solution:

1. Combine like terms: 3a + 5b = 10a
2. Subtract 3a from both sides: 5b = 7a
3. Divide both sides by 5: b = 7a/5

Problem 3: 2p - 8q + 3q = 5p

Solution:

1. Combine like terms: 2p - 5q = 5p
2. Subtract 2p from both sides: -5q = 3p
3. Divide both sides by -5: q = -3p/5

Advanced Applications and Extensions

The principles used to solve the equation 5w + 9z + 2z = 3w extend to more complex scenarios. These include:

• Systems of Equations: Solving for multiple variables simultaneously in multiple equations. Techniques like substitution or elimination are used.
• Inequalities: Equations involving inequality symbols (<, >, ≤, ≥). The solution process is similar, but care must be taken when multiplying or dividing by negative numbers.
• Quadratic Equations: Equations where the highest power of the variable is 2. These require different solution methods, such as factoring, the quadratic formula, or completing the square.

Conclusion: Mastering the Fundamentals

Solving for 'w' in the equation 5w + 9z + 2z = 3w might seem like a simple algebraic exercise. However, it lays the groundwork for understanding more complex mathematical concepts. By mastering the fundamental steps outlined here – combining like terms, using inverse operations, and applying the properties of equality – you'll be well-equipped to tackle a wide range of algebraic problems. Remember, practice is key! The more you work through similar equations, the more confident and proficient you'll become in solving for any variable. Keep practicing, and you’ll soon find that solving these types of equations becomes second nature.