Solving Proportions Finding X In 7/9 = X/(x-10)
In mathematics, proportions are a fundamental concept, expressing the equality of two ratios. Understanding how to solve proportions is crucial for various applications, from simple everyday calculations to complex scientific problems. In this comprehensive guide, we will delve into the intricacies of solving the proportion 7/9 = x/(x-10), providing a step-by-step approach and explaining the underlying principles.
Understanding Proportions
Before we dive into the specifics of solving the equation, let's clarify what a proportion is. A proportion states that two ratios are equal. A ratio, in its simplest form, is a comparison of two quantities. For instance, 7/9 represents the ratio of 7 to 9. When we say 7/9 = x/(x-10), we are asserting that the ratio 7/9 is equivalent to the ratio x/(x-10). Solving a proportion involves finding the value of the unknown variable (in this case, 'x') that makes the two ratios equal.
Setting Up the Proportion
The given proportion is 7/9 = x/(x-10). This equation tells us that the fraction 7/9 is equal to another fraction where the numerator is 'x' and the denominator is 'x-10'. The key to solving this type of equation is to understand that equivalent fractions represent the same proportional relationship. To find the value of 'x', we need to manipulate the equation in a way that isolates 'x' on one side. This typically involves using the properties of equality and cross-multiplication.
Cross-Multiplication: The Key Technique
Cross-multiplication is a powerful technique for solving proportions. It is based on the principle that if two ratios are equal, then the product of the numerator of the first ratio and the denominator of the second ratio is equal to the product of the denominator of the first ratio and the numerator of the second ratio. In other words, if a/b = c/d, then ad = bc. Applying this principle to our proportion, 7/9 = x/(x-10), we multiply 7 by (x-10) and 9 by x. This gives us the equation:
7 * (x - 10) = 9 * x
This step transforms the proportion into a linear equation, which is easier to solve. The cross-multiplication step effectively eliminates the fractions, allowing us to work with a more straightforward algebraic equation.
Expanding and Simplifying the Equation
Now that we have the equation 7 * (x - 10) = 9 * x, the next step is to expand the left side of the equation. We use the distributive property, which states that a * (b + c) = a * b + a * c. Applying this property, we multiply 7 by both 'x' and '-10':
7x - 70 = 9x
This expansion simplifies the equation, making it clearer how to isolate 'x'. The next step involves rearranging the terms so that all terms containing 'x' are on one side of the equation and the constant terms are on the other side.
Isolating the Variable 'x'
To isolate 'x', we need to move all terms containing 'x' to one side of the equation and all constant terms to the other side. Let's subtract 7x from both sides of the equation:
7x - 70 - 7x = 9x - 7x
This simplifies to:
-70 = 2x
Now, we have a simple equation where 'x' is multiplied by 2. To isolate 'x', we divide both sides of the equation by 2:
-70 / 2 = 2x / 2
This simplifies to:
x = -35
Thus, we have found the value of 'x' that satisfies the proportion 7/9 = x/(x-10). The solution is x = -35.
Verifying the Solution
It is always a good practice to verify the solution by substituting the value of 'x' back into the original proportion. This ensures that the solution is correct and that no algebraic errors were made during the solving process. Substituting x = -35 into the proportion 7/9 = x/(x-10), we get:
7/9 = -35/(-35 - 10)
Simplifying the right side:
7/9 = -35/(-45)
7/9 = 35/45
Now, we can simplify the fraction 35/45 by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
35/45 = (35 / 5) / (45 / 5)
35/45 = 7/9
Since 7/9 = 7/9, our solution x = -35 is correct. The verification step confirms that the value of 'x' we found satisfies the original proportion.
Common Mistakes and How to Avoid Them
When solving proportions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One common mistake is incorrectly applying the distributive property. For example, in the equation 7 * (x - 10) = 9 * x, students might forget to multiply 7 by both 'x' and '-10', leading to an incorrect expansion.
Another common mistake is making errors during the algebraic manipulation of the equation. This can include incorrect addition, subtraction, multiplication, or division. It is crucial to double-check each step to ensure that the algebraic operations are performed correctly. Additionally, students sometimes forget to verify their solutions, which can lead to accepting incorrect answers. Always substitute the value of 'x' back into the original proportion to confirm its validity.
To avoid these mistakes, it is essential to practice solving various types of proportion problems. Pay close attention to each step, double-check your calculations, and always verify your solutions. Understanding the underlying principles of proportions and the properties of equality is also crucial for success.
Real-World Applications of Proportions
Proportions are not just abstract mathematical concepts; they have numerous real-world applications. They are used in various fields, including cooking, construction, engineering, and finance. For instance, when scaling a recipe up or down, proportions are used to maintain the correct ratios of ingredients. In construction, proportions are used to ensure that building plans are accurately scaled and that materials are used efficiently.
Engineers use proportions in designing structures and machines, ensuring that components are correctly sized and that systems function as intended. In finance, proportions are used to calculate interest rates, investment returns, and currency exchange rates. Understanding proportions is therefore a valuable skill that can be applied in many practical situations.
Conclusion
Solving proportions is a fundamental skill in mathematics with wide-ranging applications. In this guide, we have explored the process of solving the proportion 7/9 = x/(x-10), providing a step-by-step approach that includes cross-multiplication, expanding and simplifying the equation, isolating the variable, and verifying the solution. We have also discussed common mistakes to avoid and the real-world applications of proportions.
By mastering the techniques outlined in this guide, you will be well-equipped to solve a variety of proportion problems and apply this knowledge in practical contexts. Remember, practice is key to success in mathematics, so continue to work through different types of problems to solidify your understanding.
Practice Problems
To further enhance your understanding of solving proportions, here are some practice problems:
- Solve for x: 3/5 = x/(x+4)
- Solve for y: 8/12 = (y-2)/y
- Solve for z: 4/7 = (z+3)/(z-1)
Try solving these problems on your own, and then verify your answers. This practice will help you build confidence and proficiency in solving proportions.
