Solve For \[$d\$\].$\[ \begin{aligned} 2 + 5d &= 67 \\ d &= \end{aligned} \\]

by ADMIN 79 views

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, 2 + 5d = 67, to find the value of the variable d. We will break down the solution into step-by-step instructions, making it easy to follow and understand.

Understanding Linear Equations

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, and x is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Given Equation

The given equation is 2 + 5d = 67. This is a linear equation in one variable, d. Our goal is to solve for d, which means we need to isolate the variable d on one side of the equation.

Step 1: Subtract 2 from Both Sides

To solve for d, we need to get rid of the constant term, 2, on the left side of the equation. We can do this by subtracting 2 from both sides of the equation.

2 + 5d = 67

Subtracting 2 from both sides gives us:

5d = 65

Step 2: Divide Both Sides by 5

Now that we have 5d = 65, we need to get rid of the coefficient, 5, that is multiplied by the variable d. We can do this by dividing both sides of the equation by 5.

5d = 65

Dividing both sides by 5 gives us:

d = 13

Conclusion

In this article, we solved the linear equation 2 + 5d = 67 to find the value of the variable d. We broke down the solution into two steps: subtracting 2 from both sides and dividing both sides by 5. By following these steps, we were able to isolate the variable d and find its value.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
  • To solve for a variable, you need to get rid of any constants that are multiplied by the variable.
  • When dividing both sides of an equation by a coefficient, make sure to divide both sides by the same value.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Common Mistakes

  • Not following the order of operations (PEMDAS) when solving linear equations.
  • Not getting rid of constants that are multiplied by the variable.
  • Dividing both sides of an equation by a coefficient without dividing both sides by the same value.

Conclusion

Introduction

In our previous article, we solved the linear equation 2 + 5d = 67 to find the value of the variable d. In this article, we will answer some common questions that students and professionals may have when it comes 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. 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 isolate the variable on one side of the equation. You can do this by using inverse operations, such as addition and subtraction, multiplication and division, and exponentiation and root extraction.

Q: What are some common methods for solving linear equations?

A: Some common methods for solving linear equations include:

  • Addition and Subtraction Method: This method involves adding or subtracting the same value to both sides of the equation to isolate the variable.
  • Multiplication and Division Method: This method involves multiplying or dividing both sides of the equation by the same value to isolate the variable.
  • Exponentiation and Root Extraction Method: This method involves using exponentiation and root extraction to isolate the variable.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an equation. The acronym PEMDAS stands for:

  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

Q: How do I handle fractions and decimals in linear equations?

A: When working with fractions and decimals in linear equations, you need to follow the same rules as you would with whole numbers. You can add, subtract, multiply, and divide fractions and decimals just like you would with whole numbers.

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

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

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 (PEMDAS)
  • Not getting rid of constants that are multiplied by the variable
  • Dividing both sides of an equation by a coefficient without dividing both sides by the same value

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

A: To check your work when solving linear equations, you can plug your answer back into the original equation and make sure it's true. You can also use a calculator to check your work.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve linear equations with ease. Remember to follow the order of operations (PEMDAS), get rid of constants that are multiplied by the variable, and divide both sides of an equation by a coefficient without dividing both sides by different values. With practice and patience, you will become proficient in solving linear equations and be able to apply them to real-world problems.