Select Yes Or No To Indicate Whether Each Value Of G G G Is A Solution To The Given Equation.${ \frac{2}{3} = \frac{5}{g+4} }$A. G = 5 3 G = \frac{5}{3} G = 3 5 ​ B. G = 4 3 G = \frac{4}{3} G = 3 4 ​ C. G = 7 2 G = \frac{7}{2} G = 2 7 ​

Introduction

In this article, we will explore the process of solving an equation for a variable, specifically the variable g. We will examine a given equation and determine whether each value of g is a solution to the equation. This will involve using algebraic techniques to isolate the variable g and evaluate the given expressions.

The Given Equation

The given equation is:

$\frac{2}{3} = \frac{5}{g+4}$

This equation involves a fraction on both sides, and our goal is to solve for the variable g.

Step 1: Cross-Multiplication

To begin solving the equation, we will use the technique of cross-multiplication. This involves multiplying both sides of the equation by the denominators of the fractions, which are 3 and (g+4).

$\frac{2}{3} \cdot 3 \cdot (g+4) = \frac{5}{g+4} \cdot 3 \cdot (g+4)$

This simplifies to:

$2(g+4) = 5 \cdot 3$

Step 2: Distributing and Simplifying

Next, we will distribute the 2 to the terms inside the parentheses and simplify the right-hand side of the equation.

$2g + 8 = 15$

Step 3: Isolating the Variable g

Now, we will isolate the variable g by subtracting 8 from both sides of the equation.

$2g = 7$

Step 4: Solving for g

Finally, we will solve for g by dividing both sides of the equation by 2.

$g = \frac{7}{2}$

Conclusion

In this article, we have solved the equation for g using algebraic techniques. We have cross-multiplied, distributed, and simplified the equation to isolate the variable g. The final solution is:

$g = \frac{7}{2}$

Discussion

Now that we have solved the equation, let's examine the given options to determine whether each value of g is a solution to the equation.

A. $g = \frac{5}{3}$

To determine whether this value of g is a solution, we will substitute it into the original equation.

$\frac{2}{3} = \frac{5}{\frac{5}{3}+4}$

Simplifying this expression, we get:

$\frac{2}{3} = \frac{5}{\frac{5+12}{3}}$

$\frac{2}{3} = \frac{5}{\frac{17}{3}}$

$\frac{2}{3} = \frac{15}{17}$

This is not equal to the original equation, so $g = \frac{5}{3}$ is not a solution.

B. $g = \frac{4}{3}$

To determine whether this value of g is a solution, we will substitute it into the original equation.

$\frac{2}{3} = \frac{5}{\frac{4}{3}+4}$

Simplifying this expression, we get:

$\frac{2}{3} = \frac{5}{\frac{4+12}{3}}$

$\frac{2}{3} = \frac{5}{\frac{16}{3}}$

$\frac{2}{3} = \frac{15}{16}$

This is not equal to the original equation, so $g = \frac{4}{3}$ is not a solution.

C. $g = \frac{7}{2}$

To determine whether this value of g is a solution, we will substitute it into the original equation.

$\frac{2}{3} = \frac{5}{\frac{7}{2}+4}$

Simplifying this expression, we get:

$\frac{2}{3} = \frac{5}{\frac{7+8}{2}}$

$\frac{2}{3} = \frac{5}{\frac{15}{2}}$

$\frac{2}{3} = \frac{10}{15}$

$\frac{2}{3} = \frac{2}{3}$

This is equal to the original equation, so $g = \frac{7}{2}$ is a solution.

Conclusion

In this article, we have solved the equation for g using algebraic techniques. We have cross-multiplied, distributed, and simplified the equation to isolate the variable g. The final solution is:

$g = \frac{7}{2}$

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Q: What is the given equation?

A: The given equation is:

$\frac{2}{3} = \frac{5}{g+4}$

Q: What is the goal of solving the equation?

A: The goal of solving the equation is to determine the value of the variable g.

Q: What algebraic techniques are used to solve the equation?

A: The algebraic techniques used to solve the equation include cross-multiplication, distributing, and simplifying.

Q: What is the final solution to the equation?

A: The final solution to the equation is:

$g = \frac{7}{2}$

Q: Why is $g = \frac{5}{3}$ not a solution to the equation?

A: $g = \frac{5}{3}$ is not a solution to the equation because when substituted into the original equation, it does not equal the original equation.

Q: Why is $g = \frac{4}{3}$ not a solution to the equation?

A: $g = \frac{4}{3}$ is not a solution to the equation because when substituted into the original equation, it does not equal the original equation.

Q: Why is $g = \frac{7}{2}$ a solution to the equation?

A: $g = \frac{7}{2}$ is a solution to the equation because when substituted into the original equation, it equals the original equation.

Q: What is the importance of solving the equation for g?

A: Solving the equation for g is important because it allows us to determine the value of the variable g, which can be used to solve other equations or problems.

Q: Can you provide a step-by-step guide to solving the equation for g?

A: Yes, here is a step-by-step guide to solving the equation for g:

1. Cross-multiply the equation.
2. Distribute and simplify the equation.
3. Isolate the variable g.
4. Solve for g.

Q: What are some common mistakes to avoid when solving the equation for g?

A: Some common mistakes to avoid when solving the equation for g include:

• Not cross-multiplying the equation.
• Not distributing and simplifying the equation.
• Not isolating the variable g.
• Not solving for g.

Q: Can you provide additional examples of solving equations for g?

A: Yes, here are some additional examples of solving equations for g:

• $\frac{3}{4} = \frac{2}{g+2}$
• $\frac{5}{6} = \frac{3}{g+3}$
• $\frac{2}{5} = \frac{4}{g+1}$

Conclusion

In this article, we have provided a step-by-step guide to solving the equation for g, as well as answered some frequently asked questions about the topic. We have also provided additional examples of solving equations for g.