Solve The Equation: ${ 6(4.5y - 12) = 9 }$

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, ${ 6(4.5y - 12) = 9 }$, and provide a step-by-step guide on how to approach it.

Understanding the Equation

Before we dive into solving the equation, let's break it down and understand what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, y) is 1. The equation is also a quadratic equation, as it can be rewritten in the form of $ax^2 + bx + c = 0$, where $a = 6$, $b = -72$, and $c = 9$.

Step 1: Distribute the Coefficient

The first step in solving the equation is to distribute the coefficient, which is 6, to the terms inside the parentheses. This will give us:

$${ 6(4.5y) - 6(12) = 9 }$$

Step 2: Simplify the Equation

Next, we simplify the equation by multiplying the coefficient with the terms inside the parentheses. This will give us:

$${ 27y - 72 = 9 }$$

Step 3: Add 72 to Both Sides

To isolate the term with the variable, we need to add 72 to both sides of the equation. This will give us:

$${ 27y = 81 }$$

Step 4: Divide Both Sides by 27

Finally, we divide both sides of the equation by 27 to solve for y. This will give us:

$${ y = 3 }$$

Conclusion

In this article, we solved a linear equation using a step-by-step approach. We started by distributing the coefficient, simplified the equation, added 72 to both sides, and finally divided both sides by 27 to solve for y. This equation is a great example of how to approach and solve linear equations, and we hope that this guide has been helpful in understanding the process.

Tips and Tricks

Here are some tips and tricks to keep in mind when solving linear equations:

• Distribute the coefficient: When solving an equation with parentheses, make sure to distribute the coefficient to the terms inside the parentheses.
• Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary operations.
• Add or subtract the same value to both sides: When solving an equation, make sure to add or subtract the same value to both sides to maintain the equality of the equation.
• Divide both sides by the same value: When solving an equation, make sure to divide both sides by the same value to solve for the variable.

Common Mistakes to Avoid

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

• Not distributing the coefficient: Failing to distribute the coefficient to the terms inside the parentheses can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions and make it more difficult to solve.
• Not adding or subtracting the same value to both sides: Failing to add or subtract the same value to both sides can lead to incorrect solutions and make it more difficult to solve.
• Not dividing both sides by the same value: Failing to divide both sides by the same value can lead to incorrect solutions and make it more difficult to solve.

Real-World Applications

Linear equations have many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects and the forces acting on them.
• 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 economic trends.
• Computer Science: Linear equations are used in computer algorithms and data structures to solve problems and make predictions.

Conclusion

Introduction

In our previous article, we provided a step-by-step guide on how to solve linear equations. However, we understand that sometimes, it's not enough to just follow a guide. You may have questions, and that's where this Q&A article comes in. We'll answer some of the most frequently asked questions about solving linear equations, and provide additional tips and tricks to help you master this skill.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, y) is 1. It can be written in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants.

Q: How do I know if an equation is linear or not?

A: To determine if an equation is linear or not, look for the highest power of the variable. If it's 1, then the equation is linear. If it's greater than 1, then the equation is not linear.

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.

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

A: To solve a linear equation with fractions, follow these steps:

1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
2. Simplify the equation by combining like terms.
3. Add or subtract the same value to both sides to isolate the variable.
4. Divide both sides by the coefficient of the variable to solve for it.

Q: What is the distributive property, and how do I use it to solve linear equations?

A: The distributive property is a mathematical property that states that a single operation can be distributed to multiple terms. To use it to solve linear equations, follow these steps:

1. Multiply the coefficient by each term inside the parentheses.
2. Simplify the equation by combining like terms.
3. Add or subtract the same value to both sides to isolate the variable.
4. Divide both sides by the coefficient of the variable to solve for it.

Q: How do I handle negative coefficients in linear equations?

A: When handling negative coefficients in linear equations, follow these steps:

1. Multiply both sides of the equation by the negative coefficient to eliminate the negative sign.
2. Simplify the equation by combining like terms.
3. Add or subtract the same value to both sides to isolate the variable.
4. Divide both sides by the coefficient of the variable to solve for it.

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

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

• Not distributing the coefficient to the terms inside the parentheses.
• Not simplifying the equation by combining like terms.
• Not adding or subtracting the same value to both sides to isolate the variable.
• Not dividing both sides by the coefficient of the variable to solve for it.

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

A: To check your work when solving linear equations, follow these steps:

1. Plug the solution back into the original equation to see if it's true.
2. Simplify the equation to see if it matches the original equation.
3. Check if the solution satisfies the equation.

Conclusion

In this Q&A article, we've answered some of the most frequently asked questions about solving linear equations. We've also provided additional tips and tricks to help you master this skill. Remember to always follow the steps outlined in our previous article, and to check your work to ensure that you're getting the correct solution. With practice and patience, you'll become a pro at solving linear equations in no time!