Solve For \[$ D \$\]:$\[ 3d - 3(6 - 2d) = 27 \\]

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 linear equation, which involves isolating the variable. We will use the given equation as an example and walk through the step-by-step process to solve for the variable.

The Given Equation

The given equation is:

$$3d - 3(6 - 2d) = 27$$

Our goal is to solve for the variable $d$.

Step 1: Distribute the Negative Sign

To start solving the equation, we need to distribute the negative sign inside the parentheses. This will help us simplify the equation and make it easier to work with.

$$3d - 3(6 - 2d) = 3d - 18 + 6d$$

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. In this case, we have two terms with the variable $d$, which we can combine by adding their coefficients.

$$3d - 18 + 6d = 9d - 18$$

Step 3: Add 18 to Both Sides

To isolate the variable $d$, we need to get rid of the constant term on the same side as the variable. We can do this by adding 18 to both sides of the equation.

$$9d - 18 = 27$$

$$9d = 45$$

Step 4: Divide Both Sides by 9

Now that we have isolated the variable $d$ on one side of the equation, we can divide both sides by 9 to solve for $d$.

$$d = \frac{45}{9}$$

Step 5: Simplify the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 9.

$$d = \frac{5}{1}$$

Conclusion

In this article, we have walked through the step-by-step process of solving a linear equation. We started with the given equation and used the distributive property, combined like terms, added 18 to both sides, and finally divided both sides by 9 to solve for the variable $d$. By following these steps, we were able to isolate the variable and find the solution to the equation.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
• When combining like terms, make sure to add or subtract the coefficients of the terms with the same variable.
• When adding or subtracting a constant term, make sure to add or subtract the same value from both sides of the equation.

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.

Practice Problems

To practice solving linear equations, try the following problems:

• Solve for $x$ in the equation $2x + 5 = 11$.
• Solve for $y$ in the equation $3y - 2 = 7$.
• Solve for $z$ in the equation $4z + 1 = 9$.

Conclusion

Introduction

In our previous article, we walked through the step-by-step process of solving a linear equation. However, we know that practice makes perfect, and the best way to learn is by doing. In this article, we will provide a Q&A guide to help you practice solving linear equations and address any questions you may have.

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.

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 the following steps:

1. Add or subtract the same value from both sides of the equation to get rid of any constants on the same side as the variable.
2. Use the distributive property to expand any parentheses.
3. Combine like terms by adding or subtracting the coefficients of the terms with the same variable.
4. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: What is the distributive property?

A: The distributive property is a rule that allows you to expand any parentheses by multiplying the term outside the parentheses by each term inside the parentheses. For example, if you have the expression 2(x + 3), you can use the distributive property to expand it as 2x + 6.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable. For example, if you have the expression 2x + 3x, you can combine the like terms by adding the coefficients to get 5x.

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 you have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate any expressions inside parentheses first.
2. Exponents: Evaluate any exponents 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 solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to follow the same steps as solving a linear equation with integers. However, you may need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 2 and 3 is 6.

Q: How do I solve a linear equation with decimals?

A: To solve a linear equation with decimals, you need to follow the same steps as solving a linear equation with integers. However, you may need to multiply both sides of the equation by a power of 10 to eliminate the decimals.

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 on one side of the equation
• Not checking your work

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article and practicing with the Q&A guide, you can become proficient in solving linear equations and apply the concepts to a wide range of problems. Remember to follow the order of operations, combine like terms, and isolate the variable to solve for the variable. With practice and patience, you will become a master of solving linear equations.