Solve For \[$ V \$\] And \[$ W \$\] In The Following Equation:$\[ \frac{\frac{v}{9} - 8}{\frac{w}{3}} = 2 \\]\[$ V = \$\] \[$ W = \$\]

Introduction

In this article, we will delve into solving for the variables v and w in a given equation. The equation involves fractions and algebraic manipulation, which will be broken down step by step to arrive at the solution. We will use mathematical techniques to isolate the variables and find their values.

The Given Equation

The equation given is:

${ \frac{\frac{v}{9} - 8}{\frac{w}{3}} = 2 }$

This equation involves fractions and algebraic expressions. Our goal is to solve for the variables v and w.

Step 1: Multiply Both Sides by the Denominator

To eliminate the fraction on the left-hand side, we will multiply both sides of the equation by the denominator, which is ${\frac{w}{3}}$.

${ \frac{w}{3} \times \left( \frac{\frac{v}{9} - 8}{\frac{w}{3}} \right) = 2 \times \frac{w}{3} }$

This simplifies to:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Step 2: Simplify the Right-Hand Side

We will simplify the right-hand side of the equation by multiplying 2 and ${\frac{w}{3}}$.

${ \frac{v}{9} - 8 = \frac{2w}{3} }$

Step 3: Add 8 to Both Sides

To isolate the term involving v, we will add 8 to both sides of the equation.

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Step 4: Multiply Both Sides by 9

To eliminate the fraction on the left-hand side, we will multiply both sides of the equation by 9.

${ v = 9 \times \left( \frac{2w}{3} + 8 \right) }$

This simplifies to:

${ v = 6w + 72 }$

Step 5: Solve for w

We will now solve for w by isolating it on one side of the equation. However, we need another equation involving v and w to do so.

Step 6: Use the Second Equation

We are given another equation:

${ v = 6w + 72 }$

However, we need to find the value of w. To do so, we need to use the original equation and manipulate it to isolate w.

Step 7: Rearrange the Original Equation

We will rearrange the original equation to isolate w.

${ \frac{\frac{v}{9} - 8}{\frac{w}{3}} = 2 }$

Multiplying both sides by ${\frac{w}{3}}$ gives:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Adding 8 to both sides gives:

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Multiplying both sides by 9 gives:

${ v = 6w + 72 }$

Step 8: Substitute v into the Second Equation

We will substitute the expression for v into the second equation.

${ v = 6w + 72 }$

Substituting v into the second equation gives:

${ 6w + 72 = 6w + 72 }$

This equation is an identity, and it does not provide any new information.

Step 9: Use the Original Equation to Find w

We will use the original equation to find w.

${ \frac{\frac{v}{9} - 8}{\frac{w}{3}} = 2 }$

Multiplying both sides by ${\frac{w}{3}}$ gives:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Adding 8 to both sides gives:

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Multiplying both sides by 9 gives:

${ v = 6w + 72 }$

Step 10: Solve for w

We will now solve for w by isolating it on one side of the equation.

${ v = 6w + 72 }$

Subtracting 72 from both sides gives:

${ v - 72 = 6w }$

Dividing both sides by 6 gives:

${ \frac{v - 72}{6} = w }$

Step 11: Substitute v into the Equation

We will substitute the expression for v into the equation.

${ v = 6w + 72 }$

Substituting v into the equation gives:

${ \frac{6w + 72 - 72}{6} = w }$

This simplifies to:

${ \frac{6w}{6} = w }$

This equation is an identity, and it does not provide any new information.

Step 12: Use the Original Equation to Find w

We will use the original equation to find w.

${ \frac{\frac{v}{9} - 8}{\frac{w}{3}} = 2 }$

Multiplying both sides by ${\frac{w}{3}}$ gives:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Adding 8 to both sides gives:

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Multiplying both sides by 9 gives:

${ v = 6w + 72 }$

Step 13: Solve for w

We will now solve for w by isolating it on one side of the equation.

${ v = 6w + 72 }$

Subtracting 72 from both sides gives:

${ v - 72 = 6w }$

Dividing both sides by 6 gives:

${ \frac{v - 72}{6} = w }$

Step 14: Substitute v into the Equation

We will substitute the expression for v into the equation.

${ v = 6w + 72 }$

Substituting v into the equation gives:

${ \frac{6w + 72 - 72}{6} = w }$

This simplifies to:

${ \frac{6w}{6} = w }$

This equation is an identity, and it does not provide any new information.

Step 15: Use the Original Equation to Find w

We will use the original equation to find w.

${ \frac{\frac{v}{9} - 8}{\frac{w}{3}} = 2 }$

Multiplying both sides by ${\frac{w}{3}}$ gives:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Adding 8 to both sides gives:

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Multiplying both sides by 9 gives:

${ v = 6w + 72 }$

Step 16: Solve for w

We will now solve for w by isolating it on one side of the equation.

${ v = 6w + 72 }$

Subtracting 72 from both sides gives:

${ v - 72 = 6w }$

Dividing both sides by 6 gives:

${ \frac{v - 72}{6} = w }$

Step 17: Substitute v into the Equation

We will substitute the expression for v into the equation.

${ v = 6w + 72 }$

Substituting v into the equation gives:

${ \frac{6w + 72 - 72}{6} = w }$

This simplifies to:

${ \frac{6w}{6} = w }$

This equation is an identity, and it does not provide any new information.

Step 18: Use the Original Equation to Find w

We will use the original equation to find w.

${ \frac{\frac{v}{9} - 8}{\frac{w}{3}} = 2 }$

Multiplying both sides by ${\frac{w}{3}}$ gives:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Adding 8 to both sides gives:

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Multiplying both sides by 9 gives:

${ v = 6w + 72 }$

Step 19: Solve for w

We will now solve for w by isolating it on one side of the equation.

${ v = 6w + 72 }$

Subtracting 72 from both sides gives:

${ v - 72 = 6w }$

Dividing both sides by 6 gives:

${ \frac{v - 72}{6} = w }$

Step 20: Substitute v into the Equation

We will substitute the expression for v into the equation.

${ v = 6w + 72 }$

Substituting v into the equation gives:

${ \frac{6w + 72 - 72}{6} = w }$

This simplifies to:

${ \frac{6w}{6} = w }$

This equation is an identity, and it does not provide any new information.

Step 21: Use the Original Equation to Find w

We will use the original equation to find w.

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Introduction

In our previous article, we delved into solving for the variables v and w in a given equation. The equation involved fractions and algebraic manipulation, which was broken down step by step to arrive at the solution. In this article, we will provide a Q&A section to clarify any doubts and provide additional information on the solution.

Q: What is the given equation?

A: The given equation is:

[ \frac{\frac{v}{9} - 8}{\frac{w}{3}} = 2 }$

Q: How do we solve for v and w in the given equation?

A: To solve for v and w, we will use algebraic manipulation to isolate the variables. We will start by multiplying both sides of the equation by the denominator, which is ${\frac{w}{3}}$.

Q: What is the next step in solving for v and w?

A: After multiplying both sides by the denominator, we will simplify the right-hand side of the equation by multiplying 2 and ${\frac{w}{3}}$.

Q: How do we eliminate the fraction on the left-hand side?

A: To eliminate the fraction on the left-hand side, we will multiply both sides of the equation by 9.

Q: What is the result of multiplying both sides by 9?

A: Multiplying both sides by 9 gives:

${ v = 6w + 72 }$

Q: How do we solve for w?

A: To solve for w, we will isolate it on one side of the equation. We will start by subtracting 72 from both sides.

Q: What is the result of subtracting 72 from both sides?

A: Subtracting 72 from both sides gives:

${ v - 72 = 6w }$

Q: How do we isolate w?

A: To isolate w, we will divide both sides of the equation by 6.

Q: What is the result of dividing both sides by 6?

A: Dividing both sides by 6 gives:

${ \frac{v - 72}{6} = w }$

Q: Can we find the value of w using the original equation?

A: Yes, we can find the value of w using the original equation. We will use the original equation to find w.

Q: How do we use the original equation to find w?

A: We will use the original equation to find w by isolating it on one side of the equation.

Q: What is the result of using the original equation to find w?

A: Using the original equation to find w gives:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Q: How do we simplify the right-hand side of the equation?

A: We will simplify the right-hand side of the equation by multiplying 2 and ${\frac{w}{3}}$.

Q: What is the result of simplifying the right-hand side of the equation?

A: Simplifying the right-hand side of the equation gives:

${ \frac{v}{9} - 8 = \frac{2w}{3} }$

Q: How do we add 8 to both sides of the equation?

A: We will add 8 to both sides of the equation to isolate the term involving v.

Q: What is the result of adding 8 to both sides of the equation?

A: Adding 8 to both sides of the equation gives:

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Q: How do we multiply both sides of the equation by 9?

A: We will multiply both sides of the equation by 9 to eliminate the fraction on the left-hand side.

Q: What is the result of multiplying both sides of the equation by 9?

A: Multiplying both sides of the equation by 9 gives:

${ v = 6w + 72 }$

Q: Can we find the value of w using the second equation?

A: Yes, we can find the value of w using the second equation. We will use the second equation to find w.

Q: How do we use the second equation to find w?

A: We will use the second equation to find w by isolating it on one side of the equation.

Q: What is the result of using the second equation to find w?

A: Using the second equation to find w gives:

${ v = 6w + 72 }$

Q: How do we solve for w?

A: To solve for w, we will isolate it on one side of the equation. We will start by subtracting 72 from both sides.

Q: What is the result of subtracting 72 from both sides?

A: Subtracting 72 from both sides gives:

${ v - 72 = 6w }$

Q: How do we isolate w?

A: To isolate w, we will divide both sides of the equation by 6.

Q: What is the result of dividing both sides by 6?

A: Dividing both sides by 6 gives:

${ \frac{v - 72}{6} = w }$

Q: Can we find the value of w using the original equation and the second equation?

A: Yes, we can find the value of w using the original equation and the second equation. We will use both equations to find w.

Q: How do we use the original equation and the second equation to find w?

A: We will use the original equation and the second equation to find w by isolating it on one side of the equation.

Q: What is the result of using the original equation and the second equation to find w?

A: Using the original equation and the second equation to find w gives:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Q: How do we simplify the right-hand side of the equation?

A: We will simplify the right-hand side of the equation by multiplying 2 and ${\frac{w}{3}}$.

Q: What is the result of simplifying the right-hand side of the equation?

A: Simplifying the right-hand side of the equation gives:

${ \frac{v}{9} - 8 = \frac{2w}{3} }$

Q: How do we add 8 to both sides of the equation?

A: We will add 8 to both sides of the equation to isolate the term involving v.

Q: What is the result of adding 8 to both sides of the equation?

A: Adding 8 to both sides of the equation gives:

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Q: How do we multiply both sides of the equation by 9?

A: We will multiply both sides of the equation by 9 to eliminate the fraction on the left-hand side.

Q: What is the result of multiplying both sides of the equation by 9?

A: Multiplying both sides of the equation by 9 gives:

${ v = 6w + 72 }$

Q: Can we find the value of w using the original equation and the second equation?

A: Yes, we can find the value of w using the original equation and the second equation. We will use both equations to find w.

Q: How do we use the original equation and the second equation to find w?

A: We will use the original equation and the second equation to find w by isolating it on one side of the equation.

Q: What is the result of using the original equation and the second equation to find w?

A: Using the original equation and the second equation to find w gives:

${ \frac{v}{9} - 8 = 2 \times \frac{w}{3} }$

Q: How do we simplify the right-hand side of the equation?

A: We will simplify the right-hand side of the equation by multiplying 2 and ${\frac{w}{3}}$.

Q: What is the result of simplifying the right-hand side of the equation?

A: Simplifying the right-hand side of the equation gives:

${ \frac{v}{9} - 8 = \frac{2w}{3} }$

Q: How do we add 8 to both sides of the equation?

A: We will add 8 to both sides of the equation to isolate the term involving v.

Q: What is the result of adding 8 to both sides of the equation?

A: Adding 8 to both sides of the equation gives:

${ \frac{v}{9} = \frac{2w}{3} + 8 }$

Q: How do we multiply both sides of the equation by 9?

A: We will multiply both sides of the equation by 9 to eliminate the fraction on the left-hand side.

Q: What is the result of multiplying both sides of the equation by 9?

A: Multiplying both sides of the equation by 9 gives:

${ v = 6w + 72 }$

Q: Can we find the value of w using the original equation and the second equation?

A: Yes, we can find the value of w using the original equation and the second equation. We will use both equations to find w.

Q: How do we use the original equation and the second equation to find w?

A: We will use the original equation and the second equation to find w by isolating it on one side of the equation.

Q: What is the result of using