Solving For X/y Given (5x+y)/(7x-3y) = 1/2 A Step-by-Step Guide

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In mathematics, solving for ratios and proportions is a fundamental skill. This article will delve into a specific problem where we are given a linear equation involving two variables, x and y, and our goal is to determine the ratio of x to y. This type of problem often appears in algebra and pre-calculus courses and is crucial for understanding more advanced mathematical concepts. Understanding how to manipulate equations and isolate variables is essential for solving such problems. The given equation, (5x+y)/(7x-3y) = 1/2, presents a classic scenario where we need to use algebraic techniques to find the relationship between x and y. This involves cross-multiplication, rearranging terms, and ultimately expressing the desired ratio. Let's break down the problem step by step to understand the methodology involved and arrive at the solution. This article aims to provide a comprehensive guide, ensuring that readers not only understand the solution but also grasp the underlying principles. By the end of this discussion, you should be well-equipped to tackle similar problems with confidence.

Problem Statement

We are given the equation:

5x+y7x−3y=12{\frac{5x+y}{7x-3y} = \frac{1}{2}}

The objective is to find the value of the ratio x/y. This problem falls under the category of linear equations and ratio-proportion problems, which are common in algebra. Solving for ratios often involves manipulating the given equation to isolate the variables and express them in a form that reveals their relationship. In this case, we have a fraction equal to another fraction, which suggests that cross-multiplication might be a useful first step. The challenge lies in correctly applying algebraic operations to simplify the equation and eventually arrive at an expression for x/y. The process will involve several steps, including cross-multiplication, rearranging terms, and possibly factoring. Understanding each step and the reasoning behind it is crucial for solving not just this problem, but also similar problems in the future. This article will guide you through each step in detail, making sure that you understand the underlying principles.

Solution

Step 1: Cross-Multiplication

The first step in solving the equation is to eliminate the fractions. We can do this by cross-multiplying. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This technique is based on the fundamental property of proportions: if a/b = c/d, then ad = bc. Applying this to our equation, we get:

2(5x+y)=1(7x−3y){2(5x + y) = 1(7x - 3y)}

This step simplifies the equation by removing the fractions, making it easier to work with. Cross-multiplication is a powerful tool for solving equations involving fractions and is a common technique used in algebra. It transforms a proportion into a linear equation, which can then be solved using standard algebraic methods. The resulting equation is a linear equation in two variables, x and y. The next step will involve distributing the constants and rearranging the terms to group like terms together. This will lead us closer to isolating the ratio x/y. Understanding the rationale behind each step is crucial for mastering algebraic problem-solving techniques. This initial step of cross-multiplication sets the stage for the subsequent steps and is essential for arriving at the correct solution.

Step 2: Distribute and Simplify

Next, we distribute the constants on both sides of the equation:

10x+2y=7x−3y{10x + 2y = 7x - 3y}

This step involves applying the distributive property, which states that a(b + c) = ab + ac. In this case, we multiply 2 by both terms inside the first parenthesis and 1 by both terms inside the second parenthesis. This simplifies the equation further by removing the parentheses and expressing the equation in a more expanded form. Simplifying equations is a crucial step in solving algebraic problems. It allows us to combine like terms and rearrange the equation into a form that is easier to manipulate. In this case, distributing the constants prepares the equation for the next step, which will involve grouping the x and y terms together. The goal is to isolate the variables and eventually express the ratio x/y. Each step in the solution process builds upon the previous step, and this simplification is a key step towards achieving our objective. Understanding the distributive property and its application is fundamental to algebraic manipulation.

Step 3: Rearrange the Terms

Now, we want to group the x terms and the y terms on opposite sides of the equation. To do this, we subtract 7x from both sides and subtract 2y from both sides:

10x−7x=−3y−2y{10x - 7x = -3y - 2y}

This step is a fundamental algebraic manipulation technique. By performing the same operation on both sides of the equation, we maintain the equality while rearranging the terms. The goal here is to isolate the x terms on one side and the y terms on the other side. This makes it easier to combine like terms and simplify the equation further. Rearranging terms is a common strategy in solving equations, and it's essential for isolating variables. In this case, we're moving closer to expressing the relationship between x and y. The next step will involve simplifying the equation by combining the terms on each side. Understanding how to rearrange terms effectively is a crucial skill in algebra and is used extensively in solving various types of equations. This step sets the stage for the final steps in finding the ratio x/y.

Step 4: Combine Like Terms

Combine the terms on each side:

3x=−5y{3x = -5y}

This step involves simplifying both sides of the equation by combining like terms. On the left side, we have 10x - 7x, which simplifies to 3x. On the right side, we have -3y - 2y, which simplifies to -5y. This simplification reduces the equation to a more concise form, making it easier to see the relationship between x and y. Combining like terms is a fundamental algebraic operation and is essential for solving equations. It helps to consolidate terms and simplify the equation, making it easier to isolate variables or, in this case, find the ratio between x and y. The equation 3x = -5y now represents a direct relationship between x and y, and the next step will involve manipulating this equation to express the ratio x/y. Understanding the process of combining like terms is crucial for mastering algebraic manipulations and solving equations efficiently.

Step 5: Find the Ratio x/y

To find the ratio x/y, we divide both sides of the equation by y and then by 3:

3xy=−5{\frac{3x}{y} = -5}

xy=−53{\frac{x}{y} = \frac{-5}{3}}

Therefore, the value of x/y is -5/3. This step involves isolating the ratio x/y by performing algebraic operations on both sides of the equation. First, we divide both sides by y to get the terms with x and y on opposite sides. Then, we divide both sides by 3 to isolate the ratio x/y. This process demonstrates how to manipulate equations to solve for specific ratios or variables. Finding the ratio x/y is the ultimate goal of the problem, and this step achieves that by expressing x in terms of y. The result, x/y = -5/3, represents the relationship between x and y that satisfies the original equation. Understanding how to isolate ratios and variables is a crucial skill in algebra and is applicable to a wide range of mathematical problems. This final step completes the solution and provides the answer to the problem.

Final Answer

The value of xy{\frac{x}{y}} is −53{-\frac{5}{3}}.

In this article, we solved a problem involving finding the ratio of x to y given a linear equation. We started with the equation (5x+y)/(7x-3y) = 1/2 and, through a series of algebraic manipulations, determined that x/y = -5/3. The key steps involved cross-multiplication, distributing constants, rearranging terms, combining like terms, and finally, isolating the desired ratio. Each step was explained in detail to provide a clear understanding of the problem-solving process. This type of problem is fundamental in algebra and serves as a building block for more advanced mathematical concepts. Mastering these techniques is essential for success in mathematics. The ability to manipulate equations, isolate variables, and solve for ratios is a valuable skill that can be applied to a wide range of problems. This article aimed to provide not just the solution, but also a comprehensive understanding of the underlying principles and techniques involved. By understanding these principles, you can confidently tackle similar problems and further enhance your mathematical skills. Practice is key to mastering these concepts, so try solving similar problems to reinforce your understanding.