Solve For $y$.$\[ -3 - 4y = 9 - Y \\]$y = [?\]

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill to master. In this article, we will focus on solving linear equations of the form $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. We will use the given equation $-3 - 4y = 9 - y$ as an example to demonstrate the step-by-step process of solving for $y$.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is in the form of a linear equation, where the variable $y$ is isolated on one side of the equation. The equation can be rewritten as:

$$-3 - 4y = 9 - y$$

Step 1: Add $4y$ to Both Sides

To isolate $y$, we need to get rid of the term $-4y$ on the left-hand side of the equation. We can do this by adding $4y$ to both sides of the equation. This will result in:

$$-3 = 9 - y + 4y$$

Step 2: Simplify the Right-Hand Side

Now that we have added $4y$ to both sides of the equation, we can simplify the right-hand side by combining like terms. The term $-y$ and $4y$ can be combined to form $3y$. The equation now becomes:

$$-3 = 9 + 3y$$

Step 3: Subtract 9 from Both Sides

To isolate $y$, we need to get rid of the constant term $9$ on the right-hand side of the equation. We can do this by subtracting $9$ from both sides of the equation. This will result in:

$$-12 = 3y$$

Step 4: Divide Both Sides by 3

Finally, to solve for $y$, we need to get rid of the coefficient $3$ on the right-hand side of the equation. We can do this by dividing both sides of the equation by $3$. This will result in:

$$y = -4$$

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation of the form $ax + by = c$. We used the given equation $-3 - 4y = 9 - y$ as an example to illustrate the process of isolating $y$. By following these steps, we were able to solve for $y$ and find the value of $y$ to be $-4$.

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 adding or subtracting terms, make sure to add or subtract the coefficients of the terms as well.
• When dividing both sides of the equation by a coefficient, make sure to divide the constant term by the same coefficient.

Common Mistakes to Avoid

• When solving linear equations, it's easy to get confused and make mistakes. Some common mistakes to avoid include:
• Adding or subtracting terms without adding or subtracting the coefficients.
• Dividing both sides of the equation by a coefficient without dividing the constant term by the same coefficient.
• Forgetting to simplify the right-hand side 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 the influence of forces.
• 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 economic trends.

Conclusion

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Introduction

In our previous article, we demonstrated the step-by-step process of solving linear equations of the form $ax + by = c$. In this article, we will answer some 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(s) is 1. In other words, it is an equation that can be written in the form $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

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

A: To determine if an equation is linear, look for the following characteristics:

• The highest power of the variable(s) is 1.
• The equation can be written in the form $ax + by = c$.
• The equation does not contain any squared or cubed terms.

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, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example:

• Linear equation: $2x + 3y = 5$
• Quadratic equation: $x^2 + 4x + 4 = 0$

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 canceling out any common factors.
3. Solve for the variable using the usual methods.

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

A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the underlying math and be able to solve the equation by hand. This will help you to:

• Check your work and ensure that the calculator is giving you the correct answer.
• Understand the concept of linear equations and how to solve them.
• Apply the concept of linear equations to real-world problems.

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

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

• Adding or subtracting terms without adding or subtracting the coefficients.
• Dividing both sides of the equation by a coefficient without dividing the constant term by the same coefficient.
• Forgetting to simplify the right-hand side of the equation.

Q: How do I apply linear equations to real-world problems?

A: Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under the influence of forces.
• 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 economic trends.

Conclusion

In conclusion, solving linear equations is a crucial skill to master in mathematics. By following the step-by-step process outlined in this article and answering the frequently asked questions, you can become proficient in solving linear equations and apply them to real-world problems. Remember to practice regularly and seek help when needed to ensure that you understand the concept of linear equations.

Additional Resources

• Khan Academy: Linear Equations
• Mathway: Linear Equations
• Wolfram Alpha: Linear Equations

Practice Problems

1. Solve the linear equation: $2x + 3y = 5$
2. Solve the linear equation: $x^2 + 4x + 4 = 0$
3. Solve the linear equation: $\frac{2x}{3} + \frac{3y}{4} = 5$

Answer Key

1. $x = 1$, $y = 1$
2. $x = -2$
3. $x = \frac{15}{2}$, $y = \frac{20}{3}$