Solve The Equation:$\[ \frac{x-1}{4}-\frac{2x-9}{7}=0 \\]
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Introduction
Solving equations is a fundamental concept in mathematics, and it's essential to understand how to approach them. In this article, we'll focus on solving a specific equation involving fractions. The equation is given as:
We'll break down the solution into manageable steps, making it easier to understand and follow along.
Step 1: Multiply Both Sides by the Least Common Multiple (LCM)
To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 4 and 7 is 28.
28 \times \left(\frac{x-1}{4}-\frac{2x-9}{7}\right) = 28 \times 0
Step 2: Distribute the LCM to Each Fraction
Now, we'll distribute the LCM to each fraction, which will eliminate the denominators.
7(x-1) - 4(2x-9) = 0
Step 3: Expand and Simplify the Equation
Next, we'll expand and simplify the equation by multiplying the terms inside the parentheses.
7x - 7 - 8x + 36 = 0
Step 4: Combine Like Terms
We'll combine the like terms to simplify the equation further.
-x + 29 = 0
Step 5: Isolate the Variable
Now, we'll isolate the variable x by subtracting 29 from both sides of the equation.
-x = -29
Step 6: Multiply Both Sides by -1
To solve for x, we'll multiply both sides of the equation by -1.
x = 29
Conclusion
In this article, we've solved the equation step by step. We started by multiplying both sides by the LCM, then distributed the LCM to each fraction, expanded and simplified the equation, combined like terms, isolated the variable, and finally multiplied both sides by -1 to solve for x. The solution to the equation is x = 29.
Tips and Tricks
- When solving equations involving fractions, it's essential to find the LCM of the denominators and multiply both sides by it.
- Distributing the LCM to each fraction can help eliminate the denominators.
- Combining like terms can simplify the equation and make it easier to solve.
- Isolating the variable is crucial in solving equations, and multiplying both sides by -1 can help achieve this.
Real-World Applications
Solving equations is a fundamental concept in mathematics, and it has numerous real-world applications. Some examples include:
- Physics: Solving equations is essential in physics to describe the motion of objects, forces, and energies.
- Engineering: Engineers use equations to design and optimize systems, structures, and processes.
- Economics: Economists use equations to model economic systems, understand market trends, and make predictions.
- Computer Science: Computer scientists use equations to develop algorithms, model complex systems, and optimize performance.
Final Thoughts
Solving equations is a critical skill in mathematics, and it's essential to understand how to approach them. By following the steps outlined in this article, you'll be able to solve equations involving fractions and apply the concepts to real-world problems. Remember to find the LCM, distribute it to each fraction, expand and simplify the equation, combine like terms, isolate the variable, and finally multiply both sides by -1 to solve for x. With practice and patience, you'll become proficient in solving equations and apply the concepts to various fields.
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Introduction
In our previous article, we solved the equation step by step. However, we understand that sometimes, it's not enough to just provide a solution. You might have questions, and that's where this Q&A guide comes in. We'll address some common questions related to solving equations, and provide additional tips and tricks to help you become proficient in this area.
Q: What is the least common multiple (LCM) and why is it important?
A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of solving equations, the LCM is essential because it allows us to eliminate the fractions by multiplying both sides of the equation by it. This is crucial because it simplifies the equation and makes it easier to solve.
Q: How do I find the LCM of two numbers?
A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that appears in both lists. 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) and why is it important?
A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. In the context of solving equations, the GCD is important because it helps us find the LCM. By using the formula LCM(a, b) = (a × b) / GCD(a, b), we can find the LCM of two numbers.
Q: How do I distribute the LCM to each fraction?
A: To distribute the LCM to each fraction, you need to multiply each fraction by the LCM. This will eliminate the denominators and simplify the equation. For example, if the LCM is 28, you would multiply each fraction by 28:
7(x-1) - 4(2x-9) = 28(0)
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 0 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.
Q: How do I solve a quadratic equation?
A: To solve a quadratic equation, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
Q: What is the quadratic formula and why is it important?
A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It's important because it allows us to solve quadratic equations that cannot be factored.
Q: How do I apply the quadratic formula to a quadratic equation?
A: To apply the quadratic formula to a quadratic equation, you need to identify the coefficients a, b, and c. Then, you can plug these values into the quadratic formula and simplify the expression to find the solutions.
Q: What are some common mistakes to avoid when solving equations?
A: Some common mistakes to avoid when solving equations include:
- Not finding the LCM of the denominators
- Not distributing the LCM to each fraction
- Not combining like terms
- Not isolating the variable
- Not checking the solutions
Conclusion
Solving equations is a critical skill in mathematics, and it's essential to understand how to approach them. By following the steps outlined in this article, you'll be able to solve equations involving fractions and apply the concepts to real-world problems. Remember to find the LCM, distribute it to each fraction, expand and simplify the equation, combine like terms, isolate the variable, and finally multiply both sides by -1 to solve for x. With practice and patience, you'll become proficient in solving equations and apply the concepts to various fields.
Final Thoughts
Solving equations is a complex topic, and it's essential to practice and review the concepts to become proficient. By following the steps outlined in this article, you'll be able to solve equations involving fractions and apply the concepts to real-world problems. Remember to find the LCM, distribute it to each fraction, expand and simplify the equation, combine like terms, isolate the variable, and finally multiply both sides by -1 to solve for x. With practice and patience, you'll become proficient in solving equations and apply the concepts to various fields.