Solve The Equation: ${ \frac{2x-3}{2} - \frac{3x+1}{4} = 1 }$

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Introduction

In this article, we will delve into the world of algebra and solve a complex equation step by step. The equation we will be solving is 2x−32−3x+14=1\frac{2x-3}{2} - \frac{3x+1}{4} = 1. This equation involves fractions, variables, and constants, making it a challenging problem for those who are new to algebra. However, with a clear understanding of the steps involved, we can break down the solution into manageable parts and arrive at the final answer.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its structure. The equation consists of two fractions, each with a variable and a constant in the numerator and a constant in the denominator. The equation is set equal to 1, which means we need to find the value of x that makes the equation true.

Identifying the Least Common Multiple (LCM)

To simplify the equation, we need to find the least common multiple (LCM) of the denominators, which are 2 and 4. The LCM of 2 and 4 is 4. This means we can multiply both sides of the equation by 4 to eliminate the fractions.

Step 1: Multiply Both Sides by 4

By multiplying both sides of the equation by 4, we can eliminate the fractions and simplify the equation.

2x−32⋅4−3x+14⋅4=1⋅4\frac{2x-3}{2} \cdot 4 - \frac{3x+1}{4} \cdot 4 = 1 \cdot 4

This simplifies to:

2(2x−3)−(3x+1)=42(2x-3) - (3x+1) = 4

Step 2: Distribute the Numbers

Now that we have eliminated the fractions, we can distribute the numbers inside the parentheses.

2(2x)−2(3)−3x−1=42(2x) - 2(3) - 3x - 1 = 4

This simplifies to:

4x−6−3x−1=44x - 6 - 3x - 1 = 4

Step 3: Combine Like Terms

We can now combine like terms by adding or subtracting the coefficients of the variables.

(4x−3x)−6−1=4(4x - 3x) - 6 - 1 = 4

This simplifies to:

x−7=4x - 7 = 4

Step 4: Add 7 to Both Sides

To isolate the variable x, we need to add 7 to both sides of the equation.

x−7+7=4+7x - 7 + 7 = 4 + 7

This simplifies to:

x=11x = 11

Conclusion

In this article, we have solved the equation 2x−32−3x+14=1\frac{2x-3}{2} - \frac{3x+1}{4} = 1 step by step. We started by identifying the least common multiple (LCM) of the denominators, then multiplied both sides of the equation by the LCM to eliminate the fractions. We then distributed the numbers inside the parentheses, combined like terms, and finally added 7 to both sides of the equation to isolate the variable x. The final answer is x = 11.

Final Answer

The final answer is 11\boxed{11}.

Related Topics

  • Solving linear equations
  • Simplifying fractions
  • Distributing numbers
  • Combining like terms

Further Reading

If you are interested in learning more about solving equations, I recommend checking out the following resources:

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

References

  • [1] Algebra: A Comprehensive Introduction, by Michael Artin
  • [2] College Algebra, by James Stewart
  • [3] Algebra and Trigonometry, by Michael Sullivan

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Introduction

In our previous article, we solved the equation 2x−32−3x+14=1\frac{2x-3}{2} - \frac{3x+1}{4} = 1 step by step. However, we understand that sometimes it's not enough to just provide a solution to a problem. Sometimes, you need to understand the reasoning behind the solution, and that's where a Q&A guide comes in. In this article, we'll answer some of the most frequently asked questions about solving the equation, and provide additional insights and explanations to help you better understand the solution.

Q&A

Q: What is the least common multiple (LCM) of 2 and 4?

A: The least common multiple (LCM) of 2 and 4 is 4. This means that we can multiply both sides of the equation by 4 to eliminate the fractions.

Q: Why do we need to find the LCM of the denominators?

A: We need to find the LCM of the denominators because it allows us to eliminate the fractions in the equation. By multiplying both sides of the equation by the LCM, we can simplify the equation and make it easier to solve.

Q: What is the difference between distributing numbers and combining like terms?

A: Distributing numbers involves multiplying the numbers inside the parentheses by the coefficients outside the parentheses. Combining like terms involves adding or subtracting the coefficients of the variables. In the equation 2(2x−3)−(3x+1)=42(2x-3) - (3x+1) = 4, we first distribute the numbers, and then combine like terms.

Q: Why do we need to add 7 to both sides of the equation?

A: We need to add 7 to both sides of the equation to isolate the variable x. By adding 7 to both sides, we can get rid of the constant term on the left-hand side of the equation, and solve for x.

Q: What is the final answer to the equation?

A: The final answer to the equation is x = 11.

Q: Can you provide more examples of solving equations?

A: Yes, we can provide more examples of solving equations. Here are a few examples:

  • x+23+x−14=2\frac{x+2}{3} + \frac{x-1}{4} = 2
  • 2x−15−3x+26=1\frac{2x-1}{5} - \frac{3x+2}{6} = 1
  • x+12+x−23=3\frac{x+1}{2} + \frac{x-2}{3} = 3

Q: How do I know which method to use to solve an equation?

A: The method you use to solve an equation depends on the type of equation you are working with. If the equation has fractions, you may need to use the method of finding the LCM and multiplying both sides of the equation by the LCM. If the equation has variables and constants, you may need to use the method of distributing numbers and combining like terms.

Conclusion

In this article, we've answered some of the most frequently asked questions about solving the equation 2x−32−3x+14=1\frac{2x-3}{2} - \frac{3x+1}{4} = 1. We've provided additional insights and explanations to help you better understand the solution, and we've also provided more examples of solving equations. We hope this article been helpful in your understanding of solving equations.

Final Answer

The final answer is 11\boxed{11}.

Related Topics

  • Solving linear equations
  • Simplifying fractions
  • Distributing numbers
  • Combining like terms

Further Reading

If you are interested in learning more about solving equations, I recommend checking out the following resources:

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

References

  • [1] Algebra: A Comprehensive Introduction, by Michael Artin
  • [2] College Algebra, by James Stewart
  • [3] Algebra and Trigonometry, by Michael Sullivan