Solving $5x + 10y = 18$ By Substitution A Step-by-Step Guide

Introduction

In the realm of mathematics, solving systems of equations is a fundamental skill. Often, we encounter scenarios where two or more equations intertwine, and our mission is to find the values of the variables that satisfy all equations simultaneously. Among the arsenal of techniques, substitution stands out as a powerful method. This article delves into the application of substitution to solve a specific system of equations. We'll meticulously dissect the problem, guiding you through each step of the solution process. By the end, you'll not only understand the mechanics of substitution but also grasp the underlying logic that makes it such a valuable tool in mathematical problem-solving. The core concept of substitution involves expressing one variable in terms of another from one equation and then plugging that expression into another equation. This process effectively reduces the number of variables in the second equation, making it easier to solve. The ability to solve systems of equations is not just confined to the classroom; it extends to real-world applications across various fields, including economics, engineering, and computer science. Understanding the nuances of substitution equips you with a versatile problem-solving strategy applicable in diverse scenarios. This article aims to solidify your understanding through a step-by-step approach, making it easier to tackle similar problems in the future. In the given problem, we are presented with two equations: $5x + 10y = 18$ and $x = y - 6$. The objective is to substitute the expression for $x$ from the second equation into the first equation and identify the resulting equation from the given options. By understanding the substitution method, you are not just learning to solve equations; you are honing your analytical and problem-solving skills, which are crucial in many areas of study and life. The following sections will provide a comprehensive walkthrough, ensuring you grasp every detail and can apply this method confidently.

Problem Statement

Let's first restate the problem we're tackling. We are given a system of two equations:

1. $5x + 10y = 18$
2. $x = y - 6$

The question asks: Which of the following equations is the result of substituting $y - 6$ in for $x$ in the first equation?

The options provided are:

A. $5y - 6 + 10y = 18$ B. $5x + 10y - 6 = 18$ C. $5y - 30 + 10y = 18$

The core of this problem lies in understanding how to correctly apply the substitution method. We have an expression for $x$ in terms of $y$ from the second equation, and we need to replace $x$ in the first equation with this expression. The key here is to ensure that the substitution is performed accurately, paying close attention to any distributive properties that might be involved. This type of problem is common in algebra and serves as a building block for more complex mathematical concepts. It tests your ability to manipulate equations and apply fundamental algebraic principles. The correct approach involves carefully replacing every instance of $x$ in the first equation with the expression $y - 6$. This substitution should then simplify the equation into a form that only involves the variable $y$, allowing us to solve for it if needed. However, in this case, the problem only asks for the resulting equation after the substitution, not the solution. Therefore, our primary focus is on accurate substitution and simplification. Understanding this process is crucial for solving more complex systems of equations and for applying algebraic concepts in various mathematical and real-world scenarios. The following sections will walk you through the detailed steps of this substitution process, highlighting potential pitfalls and ensuring clarity in each step.

Step-by-Step Solution

Here, we will walk through the process of substitution step by step to arrive at the correct equation.

Step 1: Identify the Equation for Substitution

We have the equation $x = y - 6$. This equation explicitly expresses $x$ in terms of $y$, making it ideal for substitution. The key to substitution is to use an equation where one variable is already isolated or easily isolatable.

Step 2: Substitute the Expression

In the first equation, $5x + 10y = 18$, we replace $x$ with the expression $y - 6$. This gives us:

$5(y - 6) + 10y = 18$

It's crucial to enclose the expression $y - 6$ in parentheses. This ensures that the multiplication by 5 is correctly applied to the entire expression, not just the first term. Omitting the parentheses is a common mistake that leads to incorrect results.

Step 3: Apply the Distributive Property

Next, we apply the distributive property to expand the term $5(y - 6)$. The distributive property states that $a(b + c) = ab + ac$. Applying this to our equation, we get:

$5 * y - 5 * 6 + 10y = 18$

This simplifies to:

$5y - 30 + 10y = 18$

The distributive property is a cornerstone of algebraic manipulation, and mastering its application is essential for accurate problem-solving. In this context, it ensures that the constant 5 is correctly multiplied across both terms within the parentheses, maintaining the equation's balance.

Step 4: Compare with the Options

Now, we compare our result, $5y - 30 + 10y = 18$, with the options provided. Option C, $5y - 30 + 10y = 18$, matches our result exactly.

Step 5: Eliminate Incorrect Options

Let's quickly examine why the other options are incorrect:

• Option A, $5y - 6 + 10y = 18$, is incorrect because it does not correctly distribute the 5 across the $(y - 6)$ term. It only subtracts 6 without multiplying it by 5.
• Option B, $5x + 10y - 6 = 18$, is incorrect because it doesn't substitute the expression for $x$ at all. It simply subtracts 6 from the left side of the original equation.

Step 6: Final Answer

Therefore, the correct equation after substituting $y - 6$ for $x$ in the first equation is:

$5y - 30 + 10y = 18$

This systematic approach to substitution, with careful attention to the distributive property, ensures accuracy and reduces the likelihood of errors. By breaking down the problem into manageable steps, we can confidently arrive at the correct solution. The ability to meticulously apply these steps is a testament to a strong foundation in algebra and problem-solving skills. This methodical approach is not only applicable to this specific problem but can be generalized to a wide range of mathematical challenges. The key takeaway here is the importance of paying attention to detail, especially when dealing with algebraic manipulations. A single error in distribution or substitution can lead to an incorrect answer. By following a structured approach, we minimize the chances of such errors and maximize our chances of success.

Correct Answer

Based on our step-by-step solution, the correct answer is:

C. $5y - 30 + 10y = 18$

This is the equation we obtained after correctly substituting $y - 6$ for $x$ in the original equation and applying the distributive property. The process we followed highlights the importance of accurate substitution and simplification in solving systems of equations. Choosing the correct answer is not just about arriving at the right result; it's about understanding the underlying principles and applying them methodically. In this case, the key was to recognize the need for the distributive property and apply it correctly. The incorrect options serve as valuable learning opportunities, illustrating common mistakes that can occur during substitution. By understanding why these options are incorrect, we reinforce our understanding of the correct method. The correct answer, option C, is a direct result of a careful and deliberate application of algebraic principles. This reinforces the idea that accuracy and precision are paramount in mathematics. The ability to solve problems like this demonstrates a solid foundation in algebraic manipulation, a skill that is crucial for further studies in mathematics and related fields. Moreover, this skill extends beyond the classroom, finding applications in various real-world scenarios where problem-solving and analytical thinking are essential.

Key Concepts Revisited

To solidify your understanding, let's revisit the key concepts involved in this problem.

Substitution Method

The substitution method is a technique used to solve systems of equations by solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be solved directly. The essence of the substitution method lies in its ability to simplify complex systems into manageable equations. By expressing one variable in terms of another, we effectively eliminate one unknown, making the problem more tractable. This method is particularly useful when one of the equations is already solved for one variable or can be easily manipulated to isolate a variable. The versatility of the substitution method makes it a fundamental tool in algebra and a stepping stone for tackling more advanced mathematical concepts. In many real-world applications, systems of equations arise naturally, and the substitution method provides a powerful means to find solutions. Whether it's in economics, engineering, or computer science, the ability to solve systems of equations is an invaluable skill.

Distributive Property

The distributive property states that $a(b + c) = ab + ac$. This property is fundamental in algebra and is used to expand expressions involving parentheses. Understanding and correctly applying the distributive property is critical for accurate algebraic manipulation. It ensures that each term within the parentheses is properly multiplied by the term outside, maintaining the integrity of the equation. The distributive property is not just a rule to be memorized; it's a fundamental concept that underpins many algebraic operations. Its applications extend far beyond simple equation solving, permeating various branches of mathematics. In the context of substitution, the distributive property often comes into play when the expression being substituted involves multiple terms. Failing to apply the distributive property correctly can lead to significant errors and incorrect solutions. Therefore, mastering this concept is essential for success in algebra and beyond.

Solving for Variables

The ultimate goal in solving systems of equations is to find the values of the variables that satisfy all equations simultaneously. Solving for variables involves isolating the variable of interest on one side of the equation, often through a series of algebraic manipulations. This process may involve addition, subtraction, multiplication, division, and the application of properties like the distributive property. The ability to solve for variables is a core skill in algebra and a prerequisite for more advanced mathematical concepts. It requires a solid understanding of algebraic principles and the ability to apply them strategically. In the context of substitution, solving for variables involves not only isolating a variable but also substituting its equivalent expression into another equation. This interplay between solving and substituting is what makes the method so powerful. The skills honed in solving for variables are not limited to mathematics; they extend to problem-solving in general, making them valuable in various fields and aspects of life.

Conclusion

In conclusion, we have successfully solved the problem by applying the substitution method and the distributive property. Understanding these concepts is crucial for tackling similar problems and building a strong foundation in algebra. The methodical approach we employed, breaking down the problem into manageable steps, is a valuable strategy for problem-solving in general. This problem serves as a microcosm of the broader challenges in mathematics and beyond. It underscores the importance of precision, attention to detail, and a systematic approach. The skills we've honed here – substitution, distribution, and solving for variables – are not just tools for answering test questions; they are fundamental skills that empower us to analyze, solve, and understand the world around us. As you continue your mathematical journey, remember that each problem is an opportunity to learn, to refine your skills, and to build confidence in your abilities. The concepts explored in this article are building blocks for more advanced topics, and a solid understanding of these basics will pave the way for future success. Keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is vast and fascinating, and the skills you acquire along the way will serve you well in countless ways.