Solve $2x - \frac{3}{4} = 7 \frac{1}{4}$A. $x = 4$ B. $x = 8$ C. $x = 3 \frac{1}{4}$ D. $x = 3 \frac{1}{2}$

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Introduction

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In this article, we will solve the linear equation $2x - \frac{3}{4} = 7 \frac{1}{4}$ using basic algebraic operations. This equation involves fractions and mixed numbers, which can be challenging to work with. However, by following a step-by-step approach, we can simplify the equation and find the value of the variable $x$.

Understanding the Equation

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The given equation is $2x - \frac{3}{4} = 7 \frac{1}{4}$. To solve this equation, we need to isolate the variable $x$ on one side of the equation. The equation involves a fraction and a mixed number, which can be converted to an improper fraction for easier calculation.

Converting the Mixed Number to an Improper Fraction

Converting $7 \frac{1}{4}$ to an Improper Fraction

To convert the mixed number $7 \frac{1}{4}$ to an improper fraction, we multiply the whole number part by the denominator and add the numerator.

$7 \frac{1}{4} = \frac{(7 \times 4) + 1}{4} = \frac{28 + 1}{4} = \frac{29}{4}$

Now that we have converted the mixed number to an improper fraction, we can rewrite the equation as:

$2x - \frac{3}{4} = \frac{29}{4}$

Simplifying the Equation

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To simplify the equation, we can start by adding $\frac{3}{4}$ to both sides of the equation. This will help us isolate the term involving $x$.

$2x - \frac{3}{4} + \frac{3}{4} = \frac{29}{4} + \frac{3}{4}$

Simplifying the left-hand side of the equation, we get:

$2x = \frac{29}{4} + \frac{3}{4}$

Adding Fractions with the Same Denominator

Adding $\frac{29}{4}$ and $\frac{3}{4}$

Since the fractions have the same denominator, we can add them by adding the numerators.

$\frac{29}{4} + \frac{3}{4} = \frac{29 + 3}{4} = \frac{32}{4}$

Now that we have added the fractions, we can simplify the equation further.

$2x = \frac{32}{4}$

Solving for $x$

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To solve for $x$, we can divide both sides of the equation by 2.

$\frac{2x}{2} = \frac{\frac{32}{4}}{2}$

Simplifying the left-hand side of the equation, we get:

$x = \frac{\frac{32}{4}}{2}$

Dividing a Fraction by a Whole Number

Dividing $\frac{32}{4}$ by 2

To divide a fraction by a whole number, we can multiply the fraction by the reciprocal of the whole number.

$\frac{\frac{32}{4}}{2} = \frac{32}{4} \times \frac{1}{2}$

Simplifying the equation, we get:

$x = \frac{32}{4} \times \frac{1}{2} = \frac{32}{8}$

Simplifying the Fraction

Simplifying $\frac{32}{8}$

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

$\frac{32}{8} = \frac{32 \div 8}{8 \div 8} = \frac{4}{1}$

Now that we have simplified the fraction, we can find the value of $x$.

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

Conclusion

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In this article, we solved the linear equation $2x - \frac{3}{4} = 7 \frac{1}{4}$ using basic algebraic operations. We converted the mixed number to an improper fraction, added fractions with the same denominator, and divided a fraction by a whole number. Finally, we simplified the fraction and found the value of $x$, which is $x = 4$.

Answer Key

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The correct answer is:

A. $x = 4$

The other options are incorrect.

Frequently Asked Questions

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Q: What is the value of $x$ in the equation $2x - \frac{3}{4} = 7 \frac{1}{4}$?

A: The value of $x$ is $x = 4$.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator.

Q: How do I add fractions with the same denominator?

A: To add fractions with the same denominator, add the numerators.

Q: How do I divide a fraction by a whole number?

A: To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number.

Q: How do I simplify a fraction?

A: To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor.

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Introduction

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In our previous article, we solved the linear equation $2x - \frac{3}{4} = 7 \frac{1}{4}$ using basic algebraic operations. We converted the mixed number to an improper fraction, added fractions with the same denominator, and divided a fraction by a whole number. In this article, we will answer some frequently asked questions related to solving linear equations with fractions and mixed numbers.

Q&A

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Q: What is the difference between a fraction and a mixed number?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, $\frac{1}{2}$ is a fraction. A mixed number is a combination of a whole number and a fraction. For example, $2 \frac{1}{2}$ is a mixed number.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator. For example, to convert $2 \frac{1}{2}$ to an improper fraction, multiply 2 by 2 and add 1:

$2 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$

Q: How do I add fractions with the same denominator?

A: To add fractions with the same denominator, add the numerators. For example, to add $\frac{1}{4}$ and $\frac{2}{4}$, add the numerators:

$\frac{1}{4} + \frac{2}{4} = \frac{1 + 2}{4} = \frac{3}{4}$

Q: How do I subtract fractions with the same denominator?

A: To subtract fractions with the same denominator, subtract the numerators. For example, to subtract $\frac{3}{4}$ from $\frac{1}{4}$, subtract the numerators:

$\frac{1}{4} - \frac{3}{4} = \frac{1 - 3}{4} = \frac{-2}{4} = -\frac{1}{2}$

Q: How do I multiply fractions?

A: To multiply fractions, multiply the numerators and multiply the denominators. For example, to multiply $\frac{1}{2}$ and $\frac{3}{4}$, multiply the numerators and multiply the denominators:

$\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}$

Q: How do I divide fractions?

A: To divide fractions, invert the second fraction and multiply. For example, to divide $\frac{1}{2}$ by $\frac{3}{4}$, invert the second fraction and multiply:

$\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3}$

Q: How do I simplify a fraction?

A: To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor. For example, to simplify $\frac{6}{8}$, divide both the numerator and the denominator by 2:

$\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$

Conclusion

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In this article, we answered some frequently asked questions related to solving linear equations with fractions and mixed numbers. We covered topics such as converting mixed numbers to improper fractions, adding and subtracting fractions with the same denominator, multiplying and dividing fractions, and simplifying fractions. We hope that this article has been helpful in clarifying any doubts you may have had about solving linear equations with fractions and mixed numbers.

Frequently Asked Questions

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Q: What is the difference between a fraction and a mixed number?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. A mixed number is a combination of a whole number and a fraction.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator.

Q: How do I add fractions with the same denominator?

A: To add fractions with the same denominator, add the numerators.

Q: How do I subtract fractions with the same denominator?

A: To subtract fractions with the same denominator, subtract the numerators.

Q: How do I multiply fractions?

A: To multiply fractions, multiply the numerators and multiply the denominators.

Q: How do I divide fractions?

A: To divide fractions, invert the second fraction and multiply.

Q: How do I simplify a fraction?

A: To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor.

Answer Key

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The correct answers to the questions are:

• Q: What is the difference between a fraction and a mixed number? A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. A mixed number is a combination of a whole number and a fraction.
• Q: How do I convert a mixed number to an improper fraction? A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator.
• Q: How do I add fractions with the same denominator? A: To add fractions with the same denominator, add the numerators.
• Q: How do I subtract fractions with the same denominator? A: To subtract fractions with the same denominator, subtract the numerators.
• Q: How do I multiply fractions? A: To multiply fractions, multiply the numerators and multiply the denominators.
• Q: How do I divide fractions? A: To divide fractions, invert the second fraction and multiply.
• Q: How do I simplify a fraction? A: To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor.