A)3x-1/5=2x+3/7b)4x/5-3x/10=2 is a set of two algebraic equations that often appear in mathematics textbooks, particularly in the context of solving linear equations with fractions. These equations are fundamental for students learning algebra because they introduce techniques such as clearing fractions, combining like terms, and isolating variables. Mastery of these types of equations enhances problem-solving skills and lays the groundwork for more advanced topics in algebra and calculus. In this comprehensive guide, we will explore both equations in detail, providing step-by-step solutions, tips, and strategies to help students and math enthusiasts understand and solve similar problems efficiently.
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Understanding the Equations
Before diving into solving these equations, it’s important to understand their structure and what they represent.
Equation A: 3x - 1/5 = 2x + 3/7
This is a linear equation involving the variable x, with fractional terms on both sides. The goal is to find the value of x that satisfies this equation.
Equation B: 4x/5 - 3x/10 = 2
Similarly, this is a linear equation with fractions, but all the fractional expressions are on the left side, and the right side is a constant.
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Strategies for Solving Linear Equations with Fractions
Fractions can complicate algebraic equations, but there are effective strategies to simplify and solve them efficiently:
- Clear Fractions: Multiply through by the least common denominator (LCD) to eliminate fractions.
- Combine Like Terms: After clearing fractions, combine similar terms to simplify the equation.
- Isolate the Variable: Use addition or subtraction to gather the variable terms on one side.
- Solve for the Variable: Divide or multiply to find the value of the variable.
- Check Your Solution: Substitute the found value back into the original equation to verify correctness.
Applying these steps systematically will make solving such equations straightforward.
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Step-by-Step Solutions
Let's now proceed to solve each equation carefully.
Solution to Equation A: 3x - 1/5 = 2x + 3/7
Step 1: Identify the Least Common Denominator (LCD)
The denominators are 5 and 7. The LCD of 5 and 7 is 35.
Step 2: Multiply both sides of the equation by the LCD to clear fractions
Multiply every term by 35:
35 (3x) - 35 (1/5) = 35 (2x) + 35 (3/7)
Calculations:
- 35 3x = 105x
- 35 (1/5) = 7
- 35 2x = 70x
- 35 (3/7) = 15
The equation becomes:
105x - 7 = 70x + 15
Step 3: Rearrange the equation to gather variable terms on one side
Subtract 70x from both sides:
105x - 70x - 7 = 15
Which simplifies to:
35x - 7 = 15
Step 4: Isolate the term with x
Add 7 to both sides:
35x = 15 + 7
35x = 22
Step 5: Solve for x
Divide both sides by 35:
x = 22 / 35
Step 6: Final answer and verification
The solution is:
x = 22/35
To verify, substitute x back into the original equation:
Left side:
3(22/35) - 1/5 = ?
Calculate:
(66/35) - 1/5
Express 1/5 as 7/35:
66/35 - 7/35 = 59/35
Right side:
2(22/35) + 3/7 = ?
Calculate:
44/35 + 3/7
Express 3/7 as 15/35:
44/35 + 15/35 = 59/35
Both sides are equal, confirming the solution is correct.
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Solution to Equation B: 4x/5 - 3x/10 = 2
Step 1: Identify the LCD of the denominators 5 and 10
The LCD of 5 and 10 is 10.
Step 2: Multiply every term by the LCD to clear fractions
Multiply both sides by 10:
10 (4x/5) - 10 (3x/10) = 10 2
Calculations:
- 10 (4x/5) = 2 4x = 8x
- 10 (3x/10) = 3x
- 10 2 = 20
The equation simplifies to:
8x - 3x = 20
Step 3: Combine like terms
8x - 3x = 5x
So,
5x = 20
Step 4: Solve for x
Divide both sides by 5:
x = 20 / 5 = 4
Step 5: Verify the solution
Substitute x = 4 into the original equation:
Left side:
(44)/5 - 34/10 = ?
Calculate:
16/5 - 12/10
Express 12/10 as 6/5:
16/5 - 6/5 = 10/5 = 2
Right side:
2
Both sides are equal, confirming the solution's correctness.
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Additional Tips for Solving Fractional Equations
While the above steps work well for these equations, here are some additional tips to make solving similar equations easier:
- Always identify the LCD first: This simplifies the process of clearing fractions.
- Double-check your arithmetic: Fractions often lead to calculation mistakes; verify each step.
- Simplify before solving: If possible, reduce fractions to their simplest form to make calculations easier.
- Be cautious with negative signs: Keep track of signs when moving terms across the equation.
- Practice with varied problems: Exposure to different types of fractional equations enhances problem-solving agility.
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Real-World Applications of Solving Equations with Fractions
Understanding how to solve equations like 3x - 1/5 = 2x + 3/7 and 4x/5 - 3x/10 = 2 is not just an academic exercise. These skills are applicable in various real-world contexts, including:
- Financial calculations: Computing interest rates, loan payments, or investment returns often involve fractional equations.
- Engineering and physics: Formulating and solving equations with ratios and proportions.
- Data analysis: Normalizing data or working with percentages and fractions.
- Statistics: Calculating probabilities and expected values that involve fractional expressions.
Mastering these algebraic techniques equips students and professionals with tools to analyze and solve complex problems across disciplines.
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Conclusion
Solving equations like 3x - 1/5 = 2x + 3/7 and 4x/5 - 3x/10 = 2 requires a strategic approach centered around clearing fractions, simplifying expressions, and isolating the variable. By understanding the underlying principles and practicing step-by-step solutions, learners can confidently tackle similar algebraic problems. Remember to verify solutions by substituting back into the original equations to ensure accuracy. With consistent practice and application of these techniques, solving fractional linear equations will become a straightforward and valuable skill in your mathematical toolkit.
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Further Resources
For those interested in deepening their understanding of algebra and fractional equations, consider exploring:
These platforms offer tutorials, practice problems, and interactive tools to enhance your learning experience.
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By mastering the techniques outlined in this guide, you'll be well-equipped to solve a variety of algebraic equations involving fractions, contributing to your overall mathematical proficiency and confidence.
