Solve For $x$: − 4 X + 4 16 = 3 + 3 ( X − 1 ) 4 -\frac{4x + 4}{16} = 3 + \frac{3(x - 1)}{4} − 16 4 X + 4 ​ = 3 + 4 3 ( X − 1 ) ​

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Introduction to Solving Linear Equations

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. In this article, we will focus on solving a specific linear equation involving fractions. The given equation is −4x+416=3+3(x−1)4-\frac{4x + 4}{16} = 3 + \frac{3(x - 1)}{4}. Our goal is to isolate the variable xx and find its value.

Understanding the Equation

Before we start solving the equation, let's break it down and understand its components. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, xx) is 1. The equation involves fractions, which can make it more challenging to solve. However, with the right approach, we can simplify the equation and solve for xx.

Step 1: Simplify the Fractions

To simplify the fractions, we need to find the least common multiple (LCM) of the denominators. In this case, the denominators are 16 and 4. The LCM of 16 and 4 is 16. We can rewrite the equation as follows:

−4x+416=3+3(x−1)4-\frac{4x + 4}{16} = 3 + \frac{3(x - 1)}{4}

−4x+416=3+3x−34-\frac{4x + 4}{16} = 3 + \frac{3x - 3}{4}

Step 2: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the LCM, which is 16. This will eliminate the fractions and make it easier to solve for xx.

−4x−4=48+12(x−1)-4x - 4 = 48 + 12(x - 1)

Step 3: Distribute the Terms

To simplify the equation, we need to distribute the terms. We can start by distributing the 12 to the terms inside the parentheses.

−4x−4=48+12x−12-4x - 4 = 48 + 12x - 12

Step 4: Combine Like Terms

Now that we have distributed the terms, we can combine like terms. We can start by combining the constant terms.

−4x−4=36+12x-4x - 4 = 36 + 12x

Step 5: Isolate the Variable

To isolate the variable xx, we need to get all the terms with xx on one side of the equation. We can start by subtracting 12x from both sides of the equation.

−16x−4=36-16x - 4 = 36

Step 6: Add 4 to Both Sides

To isolate the variable xx, we need to get rid of the constant term on the left side of the equation. We can do this by adding 4 to both sides of the equation.

−16x=40-16x = 40

Step 7: Divide Both Sides by -16

To find the value of xx, we need to divide both sides of the equation by -16.

x=−4016x = -\frac{40}{16}

Step 8: Simplify the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 8.

x=−52x = -\frac{5}{2}

Conclusion

In this article, we solved a linear equation involving fractions. We started by simplifying the fractions, eliminating the fractions, distributing the terms, combining like terms, isolating the variable, adding 4 to both sides, dividing both sides by -16, and simplifying the fraction. The final value of xx is −52-\frac{5}{2}. This is a fundamental concept in mathematics, and it is essential to understand how to solve linear equations to progress in various mathematical disciplines.

Tips and Tricks

  • When solving linear equations involving fractions, it is essential to find the least common multiple (LCM) of the denominators.
  • To eliminate the fractions, multiply both sides of the equation by the LCM.
  • Distribute the terms and combine like terms to simplify the equation.
  • Isolate the variable by getting all the terms with xx on one side of the equation.
  • Add or subtract the same value to both sides of the equation to isolate the variable.
  • Divide both sides of the equation by the coefficient of the variable to find its value.

Real-World Applications

Solving linear equations involving fractions has numerous real-world applications. For example, in physics, we use linear equations to describe the motion of objects. In economics, we use linear equations to model the behavior of markets. In engineering, we use linear equations to design and optimize systems.

Final Thoughts

Solving linear equations involving fractions is a fundamental concept in mathematics. It requires a deep understanding of algebraic concepts and techniques. By following the steps outlined in this article, you can solve linear equations involving fractions and apply them to real-world problems. Remember to find the least common multiple (LCM) of the denominators, eliminate the fractions, distribute the terms, combine like terms, isolate the variable, add or subtract the same value to both sides of the equation, and divide both sides of the equation by the coefficient of the variable to find its value.

Introduction

Solving linear equations involving fractions can be a challenging task, but with the right approach, it can be done with ease. In this article, we will answer some of the most frequently asked questions about solving linear equations involving fractions.

Q1: What is the first step in solving a linear equation involving fractions?

A1: The first step in solving a linear equation involving fractions is to find the least common multiple (LCM) of the denominators. This will help you eliminate the fractions and make it easier to solve for the variable.

Q2: How do I eliminate the fractions in a linear equation?

A2: To eliminate the fractions, multiply both sides of the equation by the LCM. This will get rid of the fractions and make it easier to solve for the variable.

Q3: What is the difference between a linear equation and a quadratic equation?

A3: A linear equation is an equation in which the highest power of the variable is 1, whereas a quadratic equation is an equation in which the highest power of the variable is 2. In this article, we are dealing with linear equations.

Q4: How do I distribute the terms in a linear equation?

A4: To distribute the terms, multiply each term inside the parentheses by the coefficient outside the parentheses. For example, if you have the equation 2(x + 3), you would multiply 2 by x and 2 by 3.

Q5: What is the purpose of combining like terms in a linear equation?

A5: Combining like terms helps to simplify the equation and make it easier to solve for the variable. Like terms are terms that have the same variable and coefficient.

Q6: How do I isolate the variable in a linear equation?

A6: To isolate the variable, get all the terms with the variable on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation.

Q7: What is the final step in solving a linear equation involving fractions?

A7: The final step in solving a linear equation involving fractions is to simplify the fraction, if necessary. This will give you the final value of the variable.

Q8: Can I use a calculator to solve a linear equation involving fractions?

A8: Yes, you can use a calculator to solve a linear equation involving fractions. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q9: What are some real-world applications of solving linear equations involving fractions?

A9: Solving linear equations involving fractions has numerous real-world applications, including physics, economics, and engineering. For example, in physics, we use linear equations to describe the motion of objects, while in economics, we use linear equations to model the behavior of markets.

Q10: How can I practice solving linear equations involving fractions?

A10: You can practice solving linear equations involving fractions by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve linear equations with fractions.

Conclusion

Solving linear equations involving fractions can be a challenging task, but with the right approach, it can be done with ease. By following the steps outlined in this article, you can solve linear equations involving fractions and apply them to real-world problems. Remember to find the least common multiple (LCM) of the denominators, eliminate the fractions, distribute the terms, combine like terms, isolate the variable, add or subtract the same value to both sides of the equation, and divide both sides of the equation by the coefficient of the variable to find its value.

Final Thoughts

Solving linear equations involving fractions is a fundamental concept in mathematics. It requires a deep understanding of algebraic concepts and techniques. By practicing and mastering the skills outlined in this article, you can become proficient in solving linear equations involving fractions and apply them to real-world problems.