Solve To Find The Value Of $x$ In The Linear Equation: $3(-4x + 5) = 12$.1. Use The Distributive Property: $\[ 3(-4x) + 3(5) = 12 \\]2. Simplify The Equation: $\[ -12x + 15 = 12 \\]3. Use The

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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, $3(-4x + 5) = 12$, using the distributive property and simplifying the equation. We will break down the solution into manageable steps, making it easy for readers to follow along.

Step 1: Use the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions with multiple terms. In this case, we have the expression $3(-4x + 5)$, and we want to use the distributive property to expand it.

${ 3(-4x + 5) = 3(-4x) + 3(5) }$

By applying the distributive property, we can see that the expression is now broken down into two separate terms: $3(-4x)$ and $3(5)$.

Step 2: Simplify the Equation

Now that we have expanded the expression using the distributive property, we can simplify the equation by combining like terms.

${ -12x + 15 = 12 }$

In this simplified equation, we can see that the term $-12x$ represents the coefficient of $x$, while the term $15$ is a constant term.

Step 3: Isolate the Variable

Our goal is to isolate the variable $x$, which means we want to get $x$ by itself on one side of the equation. To do this, we can start by subtracting $15$ from both sides of the equation.

${ -12x = 12 - 15 }$

By simplifying the right-hand side of the equation, we get:

${ -12x = -3 }$

Step 4: Solve for x

Now that we have isolated the variable $x$, we can solve for its value by dividing both sides of the equation by $-12$.

${ x = \frac{-3}{-12} }$

By simplifying the right-hand side of the equation, we get:

${ x = \frac{1}{4} }$

Conclusion

In this article, we have solved the linear equation $3(-4x + 5) = 12$ using the distributive property and simplifying the equation. We broke down the solution into manageable steps, making it easy for readers to follow along. By applying the distributive property, simplifying the equation, and isolating the variable, we were able to solve for the value of $x$.

Discussion

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we have demonstrated a step-by-step approach to solving a linear equation using the distributive property and simplifying the equation. By following these steps, readers can develop a deeper understanding of linear equations and improve their problem-solving skills.

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.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

  • Use the distributive property: The distributive property is a powerful tool for expanding expressions and simplifying equations.
  • Simplify the equation: Simplifying the equation can make it easier to solve and reduce the risk of errors.
  • Isolate the variable: Isolating the variable is a crucial step in solving linear equations.

Conclusion

Introduction

In our previous article, we explored the step-by-step process of solving a linear equation using the distributive property and simplifying the equation. In this article, we will answer some frequently asked questions about solving linear equations, providing additional insights and tips for readers to master this essential skill.

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. Linear equations 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 the highest power is 1, then the equation is linear. For example, the equation 2x + 3 = 5 is linear because the highest power of x is 1.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions with multiple terms. It states that a(b + c) = ab + ac. In the context of linear equations, the distributive property is used to expand expressions like 3(-4x + 5).

Q: How do I simplify a linear equation?

A: To simplify a linear equation, combine like terms and eliminate any parentheses. For example, the equation -12x + 15 = 12 can be simplified by combining the like terms -12x and 15.

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. For example, the equation x^2 + 4x + 4 = 0 is a quadratic equation because the highest power of x is 2.

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

A: To solve a linear equation with fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators. For example, the equation 1/2x + 1/4 = 3/4 can be solved by multiplying both sides by 4.

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

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

  • Not using the distributive property: Failing to use the distributive property can lead to incorrect solutions.
  • Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
  • Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

  • Using online resources: Websites like Khan Academy and Mathway offer interactive lessons and practice exercises.
  • Working with a tutor: A tutor can provide personalized guidance and support.
  • Solving problems in a textbook: Textbooks often include practice problems and exercises.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article and practicing regularly, readers can develop a deeper understanding of linear equations and improve their problem-solving skills. Whether you are a student or a professional, linear equations are an essential tool for solving problems and making predictions in a variety of fields.