Solve The Equation For \[$ X \$\]:$\[ \frac{x+4}{4} = \frac{2x-7}{3} \\]

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Introduction

In this article, we will delve into the world of algebra and solve a linear equation involving fractions. The equation we will be working with is x+44=2x−73\frac{x+4}{4} = \frac{2x-7}{3}. Our goal is to isolate the variable xx and find its value. We will use various techniques, such as cross-multiplication and simplification, to solve for xx.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its structure. The equation is a linear equation, meaning it is of the form ax=bax = b, where aa and bb are constants. In this case, the equation is x+44=2x−73\frac{x+4}{4} = \frac{2x-7}{3}. We can see that both sides of the equation have fractions, which means we will need to use techniques to eliminate the fractions.

Step 1: Cross-Multiply

One of the most common techniques used to solve equations involving fractions is cross-multiplication. This involves multiplying both sides of the equation by the denominators of the fractions. In this case, we will multiply both sides by 44 and 33.

x+44⋅4=2x−73⋅3\frac{x+4}{4} \cdot 4 = \frac{2x-7}{3} \cdot 3

This simplifies to:

x+4=2x−7x + 4 = 2x - 7

Step 2: Simplify the Equation

Now that we have eliminated the fractions, we can simplify the equation by combining like terms. We can subtract xx from both sides of the equation to get:

4=x−74 = x - 7

Step 3: Add 7 to Both Sides

To isolate xx, we need to get rid of the negative term on the right-hand side of the equation. We can do this by adding 77 to both sides of the equation.

4+7=x−7+74 + 7 = x - 7 + 7

This simplifies to:

11=x11 = x

Conclusion

And there you have it! We have successfully solved the equation for xx. The value of xx is 1111. We used various techniques, such as cross-multiplication and simplification, to isolate the variable xx and find its value.

Tips and Tricks

Here are a few tips and tricks to keep in mind when solving equations involving fractions:

  • Always start by cross-multiplying to eliminate the fractions.
  • Simplify the equation by combining like terms.
  • Use addition and subtraction to isolate the variable.
  • Check your work by plugging the solution back into the original equation.

Real-World Applications

Solving equations involving fractions has many real-world applications. For example, in physics, you may need to solve equations involving fractions to calculate the velocity of an object. In finance, you may need to solve equations involving fractions to calculate the interest rate on a loan.

Common Mistakes

Here are a few common mistakes to avoid when solving equations involving fractions:

  • Failing to cross-multiply, which can lead to incorrect solutions.
  • Not simplifying the equation, which can make it difficult to isolate the variable.
  • Not checking your work, which can lead to incorrect solutions.

Conclusion

Solving equations involving fractions requires a combination of techniques, including cross-multiplication and simplification. By following these steps and avoiding common mistakes, you can successfully solve equations involving fractions and find the value of the variable. Whether you're a student or a professional, solving equations involving fractions is an essential skill that can be applied to a wide range of real-world problems.

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

A: The first step in solving an equation involving fractions is to cross-multiply. This involves multiplying both sides of the equation by the denominators of the fractions.

Q: Why do we need to cross-multiply when solving equations involving fractions?

A: We need to cross-multiply because it allows us to eliminate the fractions and simplify the equation. This makes it easier to isolate the variable and find its value.

Q: What is the difference between cross-multiplication and multiplication?

A: Cross-multiplication is a specific technique used to solve equations involving fractions. It involves multiplying both sides of the equation by the denominators of the fractions. Multiplication, on the other hand, is a more general operation that can be used to solve equations involving numbers.

Q: How do I know when to use cross-multiplication versus multiplication?

A: You should use cross-multiplication when solving equations involving fractions. If the equation does not involve fractions, you can use multiplication instead.

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

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

  • Failing to cross-multiply, which can lead to incorrect solutions.
  • Not simplifying the equation, which can make it difficult to isolate the variable.
  • Not checking your work, which can lead to incorrect solutions.

Q: How do I check my work when solving equations involving fractions?

A: To check your work, plug the solution back into the original equation and simplify. If the solution is correct, the equation should be true. If the solution is incorrect, the equation will not be true.

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

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

  • Physics: Solving equations involving fractions can be used to calculate the velocity of an object.
  • Finance: Solving equations involving fractions can be used to calculate the interest rate on a loan.
  • Engineering: Solving equations involving fractions can be used to calculate the stress on a material.

Q: Can I use a calculator to solve equations involving fractions?

A: Yes, you can use a calculator to solve equations involving fractions. However, it's always a good idea to check your work by plugging the solution back into the original equation.

Q: What are some tips for solving equations involving fractions?

A: Here are some tips for solving equations involving fractions:

  • Always start by cross-multiplying to eliminate the fractions.
  • Simplify the equation by combining like terms.
  • Use addition and subtraction to isolate the variable.
  • Check your work by plugging the solution back into the original equation.

Q: Can I solve equations involving fractions with variables on both sides?

A: Yes, you can solve equations involving fractions with variables on both sides. However, it may be more difficult to isolate the variable. In this case, you may need to use additional techniques, such as adding or subtracting the same value to both sides of the equation.

Q: What are some common equations involving fractions that I should know?

A: Here are some common equations involving fractions that you should know:

  • xa=bc\frac{x}{a} = \frac{b}{c}
  • x+44=2x−73\frac{x+4}{4} = \frac{2x-7}{3}
  • x−25=x+32\frac{x-2}{5} = \frac{x+3}{2}

Q: Can I use equations involving fractions to solve problems in other areas of mathematics?

A: Yes, you can use equations involving fractions to solve problems in other areas of mathematics, such as algebra and geometry. However, the specific techniques and methods used may vary depending on the problem and the area of mathematics.