Solve For { B $} . . . { \frac{6b + 10}{4} = 7 \} { B = [?] $}$

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, $\frac{6b + 10}{4} = 7$, to find the value of the variable $b$. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Equation

The given equation is a linear equation in the form of $\frac{ax + b}{c} = d$, where $a$, $b$, $c$, and $d$ are constants. In this case, $a = 6$, $b = 10$, $c = 4$, and $d = 7$. Our goal is to isolate the variable $b$ and find its value.

Step 1: Multiply Both Sides by the Denominator

To eliminate the fraction, we need to multiply both sides of the equation by the denominator, which is $4$. This will give us:

$$6b + 10 = 7 \times 4$$

Step 2: Simplify the Right-Hand Side

Now, we simplify the right-hand side of the equation by multiplying $7$ and $4$:

$$6b + 10 = 28$$

Step 3: Subtract 10 from Both Sides

Next, we subtract $10$ from both sides of the equation to isolate the term with the variable $b$:

$$6b = 28 - 10$$

Step 4: Simplify the Right-Hand Side

We simplify the right-hand side of the equation by subtracting $10$ from $28$:

$$6b = 18$$

Step 5: Divide Both Sides by 6

Finally, we divide both sides of the equation by $6$ to solve for $b$:

$$b = \frac{18}{6}$$

Step 6: Simplify the Right-Hand Side

We simplify the right-hand side of the equation by dividing $18$ by $6$:

$$b = 3$$

Conclusion

In this article, we solved the linear equation $\frac{6b + 10}{4} = 7$ to find the value of the variable $b$. We broke down the solution into manageable steps, making it easy to understand and follow. By multiplying both sides by the denominator, simplifying the right-hand side, subtracting $10$ from both sides, simplifying the right-hand side again, dividing both sides by $6$, and simplifying the right-hand side one last time, we arrived at the solution $b = 3$. This demonstrates the importance of following a step-by-step approach when solving linear equations.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you're performing the operations in the correct order.
• When multiplying or dividing both sides of an equation by a constant, make sure to multiply or divide both sides by the same constant.
• When simplifying the right-hand side of an equation, make sure to perform the operations in the correct order.

Common Mistakes to Avoid

• When solving linear equations, it's easy to get confused and make mistakes. Some common mistakes to avoid include:
  • Multiplying or dividing both sides of an equation by a variable instead of a constant.
  • Failing to simplify the right-hand side of an equation.
  • Making errors when subtracting or adding numbers.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
• Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and structural systems.
• Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and resource allocation.

Conclusion

Introduction

In our previous article, we solved the linear equation $\frac{6b + 10}{4} = 7$ to find the value of the variable $b$. In this article, we will answer some frequently asked questions about 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. In other words, it is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable(s) on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same constant.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I handle fractions in a linear equation?

A: When a linear equation contains a fraction, you can eliminate the fraction by multiplying both sides of the equation by the denominator. For example, if the equation is $\frac{2x}{3} = 4$, you can multiply both sides by 3 to get $2x = 12$.

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. A quadratic equation, on the other hand, is an equation in which the highest power of the variable(s) is 2. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation, while the equation $2x + 3 = 5$ is a linear equation.

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 work by plugging the solution back into the original equation to make sure it's true.

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

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

• Multiplying or dividing both sides of an equation by a variable instead of a constant.
• Failing to simplify the right-hand side of an equation.
• Making errors when subtracting or adding numbers.
• Not following the order of operations.

Q: How do I check my work when solving a linear equation?

A: To check your work, plug the solution back into the original equation and make sure it's true. For example, if you solved the equation $2x + 3 = 5$ and got $x = 1$, you can plug $x = 1$ back into the original equation to get $2(1) + 3 = 5$, which is true.

Conclusion

In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following a step-by-step approach and avoiding common mistakes, you can solve linear equations with confidence. Remember to always follow the order of operations, multiply or divide both sides by the same constant, and simplify the right-hand side of an equation. With practice and patience, you'll become proficient in solving linear equations and be able to apply them to real-world problems.