Solving Equations By Substitution Substituting Y = 100 - X Into 5x + 8y = 570

In the realm of algebra, solving systems of equations is a fundamental skill. One of the most versatile and widely used techniques for tackling these systems is substitution. The substitution method allows us to solve for the unknowns in a system by expressing one variable in terms of another and then substituting that expression into another equation. This process simplifies the system, ultimately leading us to the solution. This article delves into the application of the substitution method, specifically focusing on the equation $5x + 8y = 570$ and the substitution $y = 100 - x$. We will meticulously walk through the steps involved in substituting and simplifying the equation, explaining the underlying algebraic principles at each stage. By the end of this exploration, you will have a solid grasp of how substitution works and how to confidently apply it to solve various algebraic problems.

Understanding the substitution method is crucial for students and anyone dealing with algebraic equations. It not only simplifies the process of solving equations but also deepens the understanding of how variables interact within a system. This skill is essential for higher mathematics, scientific calculations, and even everyday problem-solving scenarios. In this detailed explanation, we will break down each step, ensuring that the logic and reasoning behind the substitution process are clear. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this article provides a comprehensive guide to mastering the substitution method.

The core of the substitution method lies in replacing one variable with an equivalent expression. In this particular problem, we are given the equation $5x + 8y = 570$ and the substitution $y = 100 - x$. This means we can replace every instance of 'y' in the first equation with the expression '100 - x'. This is the foundational step that transforms the equation from one with two variables into one with a single variable, making it solvable. Let's break down this substitution process:

1. Begin with the original equation: $5x + 8y = 570$. This is the equation we aim to simplify by substituting for 'y'.
2. Identify the substitution: We have $y = 100 - x$. This equation tells us that 'y' is equivalent to the expression '100 - x'.
3. Perform the substitution: Replace 'y' in the original equation with '(100 - x)'. The equation now becomes: $5x + 8(100 - x) = 570$. This step is crucial as it eliminates one variable, allowing us to solve for the remaining one.

This substitution is not just a mechanical process; it's about understanding the relationship between the variables. By replacing 'y' with '100 - x', we are essentially incorporating the information from the second equation into the first. This transforms the problem into a simpler form that can be solved using basic algebraic techniques. The next step involves distributing and simplifying the resulting equation, which we will explore in detail in the following section. Understanding this initial substitution step is paramount, as it sets the stage for the subsequent algebraic manipulations that lead to the solution.

Following the substitution, we arrive at the equation $5x + 8(100 - x) = 570$. The next crucial step is to simplify this equation by distributing the '8' across the terms inside the parentheses. This involves multiplying '8' by both '100' and '-x'. Distribution is a key algebraic technique that allows us to remove parentheses and combine like terms, ultimately simplifying the equation. Let's walk through this process step-by-step:

1. Distribute the '8': Multiply '8' by both terms inside the parentheses: $8 * 100 = 800$ and $8 * -x = -8x$. So, the equation becomes: $5x + 800 - 8x = 570$.
2. Combine like terms: Identify terms with the same variable ('x' terms) and constant terms. In this case, we have '5x' and '-8x', which are like terms. Combining them gives us: $5x - 8x = -3x$. So, the equation now looks like: $-3x + 800 = 570$.

The distribution and simplification steps are fundamental to solving algebraic equations. They help to organize the equation and make it easier to isolate the variable we are trying to solve for. Understanding the distributive property and how to combine like terms is essential for any algebraic manipulation. In this case, by distributing the '8' and combining the 'x' terms, we have significantly simplified the equation, bringing us closer to the solution. The next step involves isolating the term with 'x' on one side of the equation, which we will delve into in the following section. This process of simplification is not just about getting to the answer; it's about developing a clear and logical approach to problem-solving in algebra.

After simplifying the equation to $-3x + 800 = 570$, our next objective is to isolate the term containing 'x', which is '-3x'. This involves moving all other terms to the opposite side of the equation. In this case, we need to eliminate the '+800' from the left side. To do this, we apply the principle of inverse operations, which states that we can perform the same operation on both sides of an equation without changing its balance. Specifically, we will subtract 800 from both sides of the equation. This step is crucial in solving for 'x' because it brings us closer to having 'x' by itself on one side of the equation. Let's break down this isolation process:

1. Subtract 800 from both sides: Start with the equation $-3x + 800 = 570$. Subtract 800 from both sides: $-3x + 800 - 800 = 570 - 800$.
2. Simplify: On the left side, '+800' and '-800' cancel each other out, leaving us with '-3x'. On the right side, 570 - 800 equals -230. So, the equation becomes: $-3x = -230$.

Isolating the variable is a cornerstone of solving algebraic equations. It demonstrates the importance of maintaining balance in an equation while manipulating it. The use of inverse operations is a fundamental technique in algebra and is applicable across a wide range of problems. In this specific case, by subtracting 800 from both sides, we have successfully isolated the term '-3x', bringing us one step closer to finding the value of 'x'. The resulting equation, $-3x = -230$, directly corresponds to one of the answer choices provided in the original problem, which we will discuss in the next section. This isolation step highlights the logical progression involved in solving equations and the power of applying basic algebraic principles.

Having simplified the equation to $-3x = -230$, the final step is to compare this result with the provided answer options to identify the correct one. This step is crucial to ensure that the algebraic manipulations we performed have led us to the accurate solution. It also reinforces the importance of carefully reviewing the work and ensuring that each step was executed correctly. In this case, the options presented are:

A. $-3x = 230$ B. $-3x = -230$ C. $-3x = 470$ D. $13x = -230$

By directly comparing our result, $-3x = -230$, with the options, it becomes clear that option B, $-3x = -230$, matches exactly. This confirms that our algebraic manipulations were accurate, and we have successfully simplified the original equation after substitution. Identifying the correct option is not just about finding a match; it's about validating the entire solution process. It's a critical step in problem-solving that ensures accuracy and builds confidence in one's algebraic skills.

This matching process also underscores the importance of attention to detail in mathematics. Even a small error in the algebraic steps can lead to an incorrect final equation. Therefore, carefully comparing the result with the given options serves as a final check, ensuring that the solution is not only logically derived but also numerically correct. In the next section, we will briefly recap the entire process, highlighting the key steps and takeaways from this problem.

In this detailed exploration, we have successfully navigated the process of substituting $y = 100 - x$ into the equation $5x + 8y = 570$ and identifying the resulting equation. Let's recap the key steps and takeaways from this problem:

1. Substitution: We began by substituting '100 - x' for 'y' in the original equation, transforming it into $5x + 8(100 - x) = 570$. This step is fundamental to the substitution method, allowing us to reduce the number of variables in the equation.
2. Distribution and Simplification: We then distributed the '8' across the terms inside the parentheses, resulting in $5x + 800 - 8x = 570$. Combining like terms, '5x' and '-8x', further simplified the equation to $-3x + 800 = 570$.
3. Isolating the Variable: To isolate the term with 'x', we subtracted 800 from both sides of the equation, yielding $-3x = -230$. This step is crucial in solving for 'x' and demonstrates the principle of maintaining balance in an equation.
4. Identifying the Correct Option: Finally, we compared our result, $-3x = -230$, with the given options and correctly identified option B as the matching equation.

The key takeaways from this problem include the importance of understanding the substitution method, the careful application of algebraic principles like distribution and combining like terms, and the need for attention to detail throughout the problem-solving process. This exercise not only demonstrates how to solve a specific algebraic problem but also reinforces fundamental skills that are essential for success in mathematics. By mastering these techniques, students and learners can confidently tackle a wide range of algebraic challenges. The ability to substitute, simplify, and isolate variables is a powerful tool in the mathematical toolkit, applicable across various contexts and disciplines.