Which Of The Following Is A Solution To This Equation?$5v + 14 = 9v + 2$A. $v = 3$ B. $v = 7$

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, $5v + 14 = 9v + 2$, and determine which of the given solutions is correct.

Understanding the Equation

The given equation is a linear equation in one variable, $v$. It can be written in the standard form as $5v + 14 - 9v = 2$. Our goal is to isolate the variable $v$ and find its value.

Step 1: Simplify the Equation

To simplify the equation, we need to combine like terms. In this case, we can combine the terms involving $v$ by subtracting $9v$ from $5v$. This gives us:

$$-4v + 14 = 2$$

Step 2: Isolate the Variable

Next, we need to isolate the variable $v$ by getting rid of the constant term on the left-hand side. We can do this by subtracting $14$ from both sides of the equation:

$$-4v = -12$$

Step 3: Solve for $v$

Now that we have isolated the variable $v$, we can solve for its value by dividing both sides of the equation by $-4$:

$$v = \frac{-12}{-4}$$

Simplifying the Expression

To simplify the expression, we can divide the numerator and denominator by their greatest common divisor, which is $-4$. This gives us:

$$v = 3$$

Evaluating the Solutions

Now that we have solved for $v$, we can evaluate the given solutions to determine which one is correct. The two solutions are:

A. $v = 3$ B. $v = 7$

Conclusion

Based on our solution, we can see that the correct value of $v$ is $3$. Therefore, the correct solution is:

The Final Answer is: A. $v = 3$

Why is this Solution Correct?

This solution is correct because we have followed the steps to solve the linear equation, and the value of $v$ we obtained is consistent with the equation. In other words, when we substitute $v = 3$ into the original equation, we get:

$$5(3) + 14 = 9(3) + 2$$

$$15 + 14 = 27 + 2$$

$$29 = 29$$

This shows that the value of $v$ we obtained is indeed a solution to the equation.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid. These include:

• Not following the order of operations
• Not combining like terms
• Not isolating the variable
• Not checking the solution by substituting it back into the equation

By avoiding these mistakes, you can ensure that your solutions are correct and accurate.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: to describe the motion of objects
• Engineering: to design and optimize systems
• Economics: to model economic systems and make predictions
• Computer Science: to solve problems and optimize algorithms

Introduction

In our previous article, we discussed how to solve a linear equation, $5v + 14 = 9v + 2$, and determined that the correct solution is $v = 3$. In this article, we will provide a Q&A guide to help you better understand how to solve linear equations and address common questions and concerns.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. In other words, it is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to follow these steps:

1. Simplify the equation by combining like terms.
2. Isolate the variable by getting rid of the constant term on the left-hand side.
3. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. In other words, a linear equation can be written in the form $ax + b = c$, while a quadratic equation can be written in the form $ax^2 + bx + c = 0$.

Q: How do I know if a solution is correct?

A: To determine if a solution is correct, you need to substitute the solution back into the original equation and check if it is true. If the solution satisfies the equation, then it is a correct solution.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

• Not following the order of operations
• Not combining like terms
• Not isolating the variable
• Not checking the solution by substituting it back into the equation

Q: How do I apply linear equations to real-world problems?

A: Linear equations have numerous real-world applications, including:

• Physics: to describe the motion of objects
• Engineering: to design and optimize systems
• Economics: to model economic systems and make predictions
• Computer Science: to solve problems and optimize algorithms

Q: Can I use linear equations to solve systems of equations?

A: Yes, you can use linear equations to solve systems of equations. A system of equations is a set of two or more linear equations that are solved simultaneously. To solve a system of equations, you need to use methods such as substitution or elimination to find the values of the variables.

Q: How do I use linear equations to model real-world problems?

A: To use linear equations to model real-world problems, you need to:

1. Identify the variables and constants in the problem.
2. Write an equation that represents the problem.
3. Solve the equation to find the values of the variables.
4. Interpret the results in the context of the problem.

Conclusion

In conclusion, solving linear equations is a crucial skill that has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can solve linear equations and apply them to real-world problems. Remember to always check your solutions by substituting them back into the original equation and to use linear equations to model real-world problems.

Additional Resources

For more information on solving linear equations and applying them to real-world problems, check out the following resources:

• Khan Academy: Linear Equations
• Mathway: Linear Equations
• Wolfram Alpha: Linear Equations

Practice Problems

Try solving the following linear equations:

1. $2x + 3 = 5x - 2$
2. $x - 2 = 3x + 1$
3. $4x + 2 = 2x - 3$

Answer Key

1. $x = -1$
2. $x = -3$
3. $x = -5/2$