Solve The Equation $-4(8 + Y) = 90$.

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, $-4(8 + y) = 90$. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Equation

The given equation is $-4(8 + y) = 90$. To solve for $y$, we need to isolate the variable on one side of the equation. The equation involves a negative coefficient, which can be challenging to work with. However, with a clear understanding of the equation and a step-by-step approach, we can easily solve for $y$.

Step 1: Distribute the Negative Coefficient

The first step in solving the equation is to distribute the negative coefficient, $-4$, to the terms inside the parentheses. This will help us simplify the equation and make it easier to work with.

-4(8 + y) = -4(8) - 4y

Step 2: Simplify the Equation

Now that we have distributed the negative coefficient, we can simplify the equation by evaluating the expression $-4(8)$.

-4(8 + y) = -32 - 4y

Step 3: Isolate the Variable

The next step is to isolate the variable, $y$, on one side of the equation. To do this, we need to get rid of the constant term, $-32$, by adding it to both sides of the equation.

-32 - 4y = 90

Step 4: Add 4y to Both Sides

Now that we have added $-32$ to both sides of the equation, we can add $4y$ to both sides to isolate the variable.

-32 = 90 + 4y

Step 5: Subtract 90 from Both Sides

The next step is to subtract $90$ from both sides of the equation to get rid of the constant term.

-122 = 4y

Step 6: Divide Both Sides by 4

Finally, we can divide both sides of the equation by $4$ to solve for $y$.

y = -122/4

Conclusion

Solving the equation $-4(8 + y) = 90$ requires a step-by-step approach. By distributing the negative coefficient, simplifying the equation, isolating the variable, and performing the necessary operations, we can easily solve for $y$. The solution to the equation is $y = -30.5$. This article has provided a clear and concise guide to solving linear equations, making it easy for students to understand and master this crucial skill.

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 working with negative coefficients, it's crucial to distribute them correctly to avoid errors.
• Isolating the variable on one side of the equation is a critical step in solving linear equations. Make sure to get rid of any constant terms by adding or subtracting them from both sides of the equation.

Common Mistakes to Avoid

• Failing to distribute the negative coefficient correctly can lead to errors in the solution.
• Not isolating the variable on one side of the equation can make it difficult to solve for the variable.
• Not following the order of operations (PEMDAS) can result in incorrect solutions.

Real-World Applications

Solving linear equations has numerous real-world applications. In physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the behavior of markets. In engineering, linear equations are used to design and optimize systems. By mastering the skill of solving linear equations, students can apply it to a wide range of real-world problems.

Practice Problems

Solving linear equations requires practice to master. Here are a few practice problems to help you improve your skills:

1. Solve the equation $2x + 5 = 11$.
2. Solve the equation $x - 3 = 7$.
3. Solve the equation $4x + 2 = 18$.

Conclusion

Introduction

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. In this article, we will provide a Q&A guide to help students and teachers clarify any doubts they may have 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 (usually x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

1. Distribute the negative coefficient (if any) to the terms inside the parentheses.
2. Simplify the equation by evaluating any expressions.
3. Isolate the variable on one side of the equation by adding or subtracting the same value to both sides.
4. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

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

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions 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 handle negative coefficients?

A: When working with negative coefficients, it's essential to distribute them correctly to avoid errors. To distribute a negative coefficient, multiply each term inside the parentheses by the negative coefficient.

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

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

• Failing to distribute the negative coefficient correctly.
• Not isolating the variable on one side of the equation.
• Not following the order of operations (PEMDAS).
• Not checking the solution to ensure it satisfies the original equation.

Q: How do I check my solution to ensure it satisfies the original equation?

A: To check your solution, substitute the value of the variable back into the original equation and evaluate it. If the equation is true, then your solution is correct.

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

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

• Physics: Linear equations are used to describe the motion of objects.
• Economics: Linear equations are used to model the behavior of markets.
• Engineering: Linear equations are used to design and optimize systems.

Q: How can I practice solving linear equations?

A: There are many resources available to help you practice solving linear equations, including:

• Online practice problems and quizzes.
• Math textbooks and workbooks.
• Online math resources and tutorials.
• Working with a tutor or teacher.

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. By following the steps outlined in this Q&A guide, you can clarify any doubts you may have about solving linear equations and improve your skills in this area. Remember to practice regularly to become proficient in solving linear equations.