Solve The Equation: $ -2(4 + 3y) = -2(4 + Y) $

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, $-2(4 + 3y) = -2(4 + y)$, 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 $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Step 1: Distribute the Negative 2

The first step in solving the equation is to distribute the negative 2 to both terms inside the parentheses. This will give us:

$$-2(4 + 3y) = -2(4 + y)$$

$$-8 - 6y = -8 - 2y$$

Step 2: Add 8 to Both Sides

The next step is to add 8 to both sides of the equation to get rid of the negative term. This will give us:

$$-8 - 6y + 8 = -8 - 2y + 8$$

$$-6y = -2y$$

Step 3: Add 2y to Both Sides

The next step is to add 2y to both sides of the equation to get rid of the negative term. This will give us:

$$-6y + 2y = -2y + 2y$$

$$-4y = 0$$

Step 4: Divide Both Sides by -4

The final step is to divide both sides of the equation by -4 to solve for y. This will give us:

$$\frac{-4y}{-4} = \frac{0}{-4}$$

$$y = 0$$

Conclusion

In this article, we have solved the linear equation $-2(4 + 3y) = -2(4 + y)$ using a step-by-step approach. We have distributed the negative 2, added 8 to both sides, added 2y to both sides, and finally divided both sides by -4 to solve for y. The solution to the equation is y = 0.

Real-World Applications

Solving linear equations is a crucial skill in many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects, including the position, velocity, and acceleration of an object.
• Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
• Economics: Linear equations are used to model economic systems, including supply and demand curves, and to make predictions about economic outcomes.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

• Use the distributive property: The distributive property states that a(b + c) = ab + ac. This can be used to simplify linear equations and make them easier to solve.
• Add or subtract the same value to both sides: Adding or subtracting the same value to both sides of a linear equation can help to isolate the variable and solve for it.
• Use inverse operations: Inverse operations, such as addition and subtraction, can be used to solve linear equations. For example, if we have the equation 2x + 3 = 5, we can subtract 3 from both sides to get 2x = 2, and then divide both sides by 2 to get x = 1.

Common Mistakes

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

• Not distributing the negative sign: Failing to distribute the negative sign can lead to incorrect solutions.
• 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.
• Not using inverse operations: Failing to use inverse operations can lead to incorrect solutions.

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. In this article, we'll answer some of the most 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 (in this case, y) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, 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 quadratic or higher.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that a(b + c) = ab + ac. This can be used to simplify linear equations and make them easier to solve.

Q: How do I use the distributive property to solve a linear equation?

A: To use the distributive property to solve a linear equation, follow these steps:

1. Distribute the negative sign to both terms inside the parentheses.
2. Add or subtract the same value to both sides to isolate the variable.
3. Use inverse operations to solve for the variable.

Q: What is an inverse operation?

A: An inverse operation is a mathematical operation that "reverses" another operation. For example, addition and subtraction are inverse operations, as are multiplication and division.

Q: How do I use inverse operations to solve a linear equation?

A: To use inverse operations to solve a linear equation, follow these steps:

1. Identify the inverse operation needed to solve the equation.
2. Apply the inverse operation to both sides of the equation.
3. Simplify the equation to solve for the variable.

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 negative sign.
• Not adding or subtracting the same value to both sides.
• Not using inverse operations.

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

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

1. Plug in the solution into the original equation.
2. Simplify the equation to see if it's true.
3. If it's true, then your solution is correct.

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

A: Solving linear equations has many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects, including the position, velocity, and acceleration of an object.
• Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
• Economics: Linear equations are used to model economic systems, including supply and demand curves, and to make predictions about economic outcomes.

Conclusion

In conclusion, solving linear equations is a crucial skill in mathematics and has many real-world applications. By following the step-by-step guide outlined in this article, you can solve linear equations with ease. Remember to use the distributive property, add or subtract the same value to both sides, and use inverse operations to solve for the variable. With practice and patience, you can become proficient in solving linear equations and apply this skill to a wide range of problems.