Solve For $a$.$\[ 7a - 2b = 5a + B \\]A. $a = 2b$ B. $a = 3b$ C. $a = \frac{3}{2}b$ D. $a = \frac{2}{3}b$

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, which involves isolating the variable $a$ in the given equation. We will use a step-by-step approach to solve for $a$ and provide a clear explanation of the process.

The Given Equation

The given equation is:

${ 7a - 2b = 5a + b }$

Our goal is to isolate the variable $a$ and express it in terms of $b$.

Step 1: Add or Subtract to Isolate the Variable

To isolate the variable $a$, we need to get all the terms involving $a$ on one side of the equation. We can do this by adding or subtracting the same value to both sides of the equation. In this case, we can add $2b$ to both sides of the equation to get:

${ 7a - 2b + 2b = 5a + b + 2b }$

This simplifies to:

${ 7a = 5a + 3b }$

Step 2: Subtract to Isolate the Variable

Next, we need to get all the terms involving $a$ on one side of the equation. We can do this by subtracting $5a$ from both sides of the equation. This gives us:

${ 7a - 5a = 5a + 3b - 5a }$

This simplifies to:

${ 2a = 3b }$

Step 3: Divide to Isolate the Variable

Finally, we need to isolate the variable $a$ by dividing both sides of the equation by 2. This gives us:

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

This simplifies to:

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

Conclusion

In this article, we have solved a linear equation to isolate the variable $a$ in terms of $b$. We used a step-by-step approach to add or subtract to isolate the variable, and then divide to get the final solution. The solution is:

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

This is the correct answer, and it matches option C in the given multiple-choice question.

Why is this Important?

Solving linear equations is an essential skill in mathematics, and it has many real-world applications. For example, in physics, linear equations are used to describe the motion of objects, while in economics, they are used to model the behavior of markets. In computer science, linear equations are used to solve systems of equations and optimize algorithms.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

• Always start by isolating the variable you want to solve for.
• Use addition and subtraction to get all the terms involving the variable on one side of the equation.
• Use division to isolate the variable.
• Check your solution by plugging it back into the original equation.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

• Not isolating the variable you want to solve for.
• Not using addition and subtraction to get all the terms involving the variable on one side of the equation.
• Not using division to isolate the variable.
• Not checking your solution by plugging it back into the original equation.

Real-World Applications

Linear equations have many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects, such as the trajectory of a projectile or the motion of a pendulum.
• Economics: Linear equations are used to model the behavior of markets, such as the supply and demand curves.
• Computer Science: Linear equations are used to solve systems of equations and optimize algorithms.

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations by isolating the variable $a$ in terms of $b$. In this article, we will provide a Q&A guide to help you understand the concept better and answer common questions related to solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:

${ ax + by = c }$

where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable(s) by using addition, subtraction, multiplication, and division. Here are the steps:

1. Simplify the equation by combining like terms.
2. Add or subtract the same value to both sides of the equation to isolate the variable(s).
3. Multiply or divide both sides of the equation by the same value to isolate the variable(s).
4. Check your solution by plugging it back into the original equation.

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(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example:

Linear equation: $2x + 3y = 5$ Quadratic equation: $x^2 + 2x + 1 = 0$

Q: Can I use a calculator to solve a linear equation?

A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your solution by plugging it back into the original equation to ensure that it's correct.

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

A: Here are some common mistakes to avoid when solving linear equations:

• Not isolating the variable(s) you want to solve for.
• Not using addition and subtraction to get all the terms involving the variable(s) on one side of the equation.
• Not using multiplication and division to isolate the variable(s).
• Not checking your solution by plugging it back into the original equation.

Q: Can I use linear equations to solve real-world problems?

A: Yes, linear equations can be used to solve real-world problems. For example, in physics, linear equations are used to describe the motion of objects, while in economics, they are used to model the behavior of markets.

Q: What are some examples of linear equations in real-world applications?

A: Here are some examples of linear equations in real-world applications:

• Physics: $v = u + at$ (equation of motion)
• Economics: $P = MC + F$ (equation of supply and demand)
• Computer Science: $y = mx + b$ (equation of a line)

Conclusion

In conclusion, solving linear equations is an essential skill in mathematics, and it has many real-world applications. By following a step-by-step approach and using addition, subtraction, multiplication, and division, you can isolate the variable(s) and solve the equation. Remember to check your solution by plugging it back into the original equation, and avoid common mistakes such as not isolating the variable(s) or not using multiplication and division. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

Additional Resources

If you want to learn more about solving linear equations, here are some additional resources:

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

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving linear equations:

1. Solve the equation: $2x + 3y = 5$
2. Solve the equation: $x^2 + 2x + 1 = 0$
3. Solve the equation: $v = u + at$
4. Solve the equation: $P = MC + F$
5. Solve the equation: $y = mx + b$

Answer Key

Here are the answers to the practice problems:

1. $x = 1$, $y = 1$
2. $x = -1$
3. $v = 10 m/s$
4. $P = 100$
5. $y = 2x + 1$