Select The Correct Answer.What Is The Solution For $x$ In The Equation? − X + 3 7 = 2 X − 25 7 -x+\frac{3}{7}=2 X-\frac{25}{7} − X + 7 3 ​ = 2 X − 7 25 ​ A. X = − 3 4 X=-\frac{3}{4} X = − 4 3 ​ B. X = 3 4 X=\frac{3}{4} X = 4 3 ​ C. X = 4 3 X=\frac{4}{3} X = 3 4 ​ D. X = − 4 3 X=-\frac{4}{3} X = − 3 4 ​

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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 type of linear equation, which involves fractions. We will use the given equation: −x+37=2x−257-x+\frac{3}{7}=2 x-\frac{25}{7}, and we will guide you through the step-by-step process of solving for xx.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at it. The equation is: −x+37=2x−257-x+\frac{3}{7}=2 x-\frac{25}{7}. We can see that it involves fractions, which can make it a bit more challenging to solve. However, with the right approach, we can simplify the equation and find the value of xx.

Step 1: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 7 is 7. So, we multiply both sides of the equation by 7:

7(−x+37)=7(2x−257)7(-x+\frac{3}{7})=7(2 x-\frac{25}{7})

This simplifies to:

−7x+3=14x−25-7x+3=14x-25

Step 2: Isolate the Variable

Now that we have eliminated the fractions, we can isolate the variable xx. To do this, we can add 7x7x to both sides of the equation:

−7x+7x+3=14x−7x−25-7x+7x+3=14x-7x-25

This simplifies to:

3=7x−253=7x-25

Step 3: Add 25 to Both Sides

Next, we can add 25 to both sides of the equation to get rid of the negative term:

3+25=7x−25+253+25=7x-25+25

This simplifies to:

28=7x28=7x

Step 4: Divide Both Sides by 7

Finally, we can divide both sides of the equation by 7 to solve for xx:

287=7x7\frac{28}{7}=\frac{7x}{7}

This simplifies to:

4=x4=x

Conclusion

In this article, we have solved the equation −x+37=2x−257-x+\frac{3}{7}=2 x-\frac{25}{7} using a step-by-step approach. We eliminated the fractions, isolated the variable, added 25 to both sides, and finally divided both sides by 7 to solve for xx. The correct answer is:

  • A. x=−34x=-\frac{3}{4}: This is incorrect.
  • B. x=34x=\frac{3}{4}: This is incorrect.
  • C. x=43x=\frac{4}{3}: This is incorrect.
  • D. x=−43x=-\frac{4}{3}: This is incorrect.
  • The correct answer is x=4x=4, but it is not listed in the options.

Discussion

This equation may seem challenging at first, but with the right approach, it can be solved easily. The key is to eliminate the fractions and isolate the variable. By following these steps, you can solve any linear equation involving fractions.

Tips and Tricks

  • When solving linear equations involving fractions, it's essential to eliminate the fractions first.
  • Use the least common multiple (LCM) to eliminate the fractions.
  • Isolate the variable by adding or subtracting the same value from both sides of the equation.
  • Finally, divide both sides of the equation by the coefficient of the variable to solve for xx.

Introduction

In our previous article, we solved the equation −x+37=2x−257-x+\frac{3}{7}=2 x-\frac{25}{7} using a step-by-step approach. We eliminated the fractions, isolated the variable, added 25 to both sides, and finally divided both sides by 7 to solve for xx. In this article, we will answer some frequently asked questions about solving linear equations involving fractions.

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

A: The first step in solving a linear equation involving fractions is to eliminate the fractions. You can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: How do I find the least common multiple (LCM) of the denominators?

A: To find the LCM of the denominators, you can list the multiples of each denominator and find the smallest multiple that is common to both. Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I isolate the variable in a linear equation involving fractions?

A: To isolate the variable in a linear equation involving fractions, you can add or subtract the same value from both sides of the equation. This will help you get rid of the fractions and isolate the variable.

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

A: The final step in solving a linear equation involving fractions is to divide both sides of the equation by the coefficient of the variable. This will give you the value of the variable.

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

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

  • Not eliminating the fractions first
  • Not isolating the variable correctly
  • Not dividing both sides of the equation by the coefficient of the variable
  • Not checking the solution for extraneous solutions

Q: How can I check if my solution is correct?

A: To check if your solution is correct, you can plug the value of the variable back into the original equation and simplify. If the equation is true, then your solution is correct.

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

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

  • Finance: Solving linear equations involving fractions can help you calculate interest rates, investment returns, and other financial metrics.
  • Science: Solving linear equations involving fractions can help you model real-world phenomena, such as the motion of objects or the behavior of chemical reactions.
  • Engineering: Solving linear equations involving fractions can help you design and optimize systems, such as electrical circuits or mechanical systems.

Conclusion

Solving linear equations involving fractions can seem challenging at first, but with the right approach, it can be done easily. By following the steps outlined in this article, you can become proficient in solving linear equations involving fractions. Remember to eliminate the fractions first, isolate the variable correctly, and divide both sides of the equation by the coefficient of the variable. With practice and patience, you can master this skill and apply it to real-world problems.