Solving Systems Of Equations By Substitution A Step-by-Step Guide

In mathematics, solving systems of equations is a fundamental skill with applications in various fields, including engineering, economics, and computer science. One of the most common methods for solving these systems is the substitution method. This article will delve into the substitution method, providing a step-by-step guide on how to apply it effectively. We will use the given system of equations as an example and explore the different scenarios that can arise, such as having no solution or infinite solutions. Understanding these concepts is crucial for mastering algebra and its applications.

Understanding Systems of Equations

Before diving into the substitution method, it's essential to understand what a system of equations is. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. Graphically, the solution represents the point(s) where the lines or curves represented by the equations intersect. There are several methods to solve systems of equations, including substitution, elimination, and graphing. Each method has its strengths and is suitable for different types of systems.

The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. This makes it straightforward to substitute this expression into the other equation, thereby reducing the system to a single equation with one variable. This equation can then be solved, and the value obtained can be substituted back to find the value of the other variable. This process, while seemingly simple, requires a clear understanding of algebraic manipulation and a systematic approach to avoid errors. In the following sections, we will explore this method in detail, using a specific example to illustrate each step.

The ability to solve systems of equations is not just a theoretical exercise; it has practical implications in numerous real-world scenarios. For instance, in economics, it can be used to determine the equilibrium price and quantity in a market. In engineering, it can help in designing circuits or structures by ensuring that various constraints are met. Computer scientists use it in algorithm design and optimization problems. Therefore, mastering the techniques for solving systems of equations, such as the substitution method, is an invaluable skill for anyone pursuing studies or careers in these fields. The elegance of the substitution method lies in its logical simplicity and its applicability to a wide range of problems, making it a cornerstone of algebraic problem-solving.

The Substitution Method: A Step-by-Step Guide

The substitution method is a powerful technique for solving systems of equations. This method involves solving one equation for one variable and substituting that expression into the other equation. This eliminates one variable, allowing you to solve for the remaining variable. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. Let's break down the steps involved in the substitution method:

1. Solve one equation for one variable: Choose one of the equations and solve it for one variable in terms of the other. This step aims to isolate one variable on one side of the equation. Look for an equation where a variable has a coefficient of 1 or -1, as this will simplify the algebraic manipulation. For instance, if you have the equations $x + y = 5$ and $2x - y = 1$, solving the first equation for $x$ (i.e., $x = 5 - y$) is a good starting point. This step sets the foundation for the substitution process, making the subsequent steps more manageable.
2. Substitute the expression into the other equation: Take the expression you found in step 1 and substitute it into the other equation. This will result in a new equation with only one variable. For example, if you solved the first equation for $x$ and obtained $x = 5 - y$, substitute this expression for $x$ in the second equation $2x - y = 1$. This substitution yields $2(5 - y) - y = 1$, which is an equation in terms of $y$ only. This step is crucial as it reduces the system to a single equation that can be solved directly.
3. Solve the resulting equation: Solve the equation obtained in step 2 for the remaining variable. This step involves standard algebraic techniques such as combining like terms, distributing, and isolating the variable. In our example, the equation $2(5 - y) - y = 1$ simplifies to $10 - 2y - y = 1$, which further simplifies to $10 - 3y = 1$. Solving for $y$ involves subtracting 10 from both sides to get $-3y = -9$, and then dividing by -3 to find $y = 3$. This step provides the value of one variable, which is a significant milestone in solving the system.
4. Substitute back to find the other variable: Substitute the value you found in step 3 back into either of the original equations or the expression you found in step 1 to solve for the other variable. It's often easier to substitute into the expression from step 1 since the variable is already isolated. For instance, if you found $y = 3$ and the expression from step 1 is $x = 5 - y$, substituting $y = 3$ gives $x = 5 - 3$, so $x = 2$. This step completes the process of finding the values of both variables.
5. Check your solution: Substitute both values back into the original equations to verify that they satisfy both equations. This is a crucial step to ensure that you haven't made any errors during the process. If the values satisfy both equations, then you have found the correct solution. If not, you need to review your steps and identify any mistakes. Checking the solution provides confidence in the answer and reinforces the understanding of the system of equations.

By following these steps methodically, you can effectively solve a wide range of systems of equations using the substitution method. Each step is designed to simplify the problem, making it more manageable and less prone to errors. The substitution method is not only a valuable tool in mathematics but also a fundamental concept that underpins various problem-solving approaches in science and engineering. Its logical structure and step-by-step nature make it an accessible and powerful technique for anyone looking to master algebraic problem-solving.

Applying Substitution to the Given System

Now, let's apply the substitution method to the given system of equations:

$egin{array}{c} 4x - y = 7 \ 3y - 12x = -21

\end{array}$

1. Solve one equation for one variable: We can start by solving the first equation for y. This is a straightforward step because the coefficient of y is -1, making it easy to isolate. Starting with the first equation, $4x - y = 7$, we can rearrange it to solve for y. Subtracting $4x$ from both sides gives $-y = 7 - 4x$. To solve for y, we multiply both sides by -1, which yields $y = 4x - 7$. This expression now gives us y in terms of x, which is a crucial step for the substitution method. Choosing the right variable to solve for initially can significantly simplify the subsequent steps. In this case, solving for y in the first equation was a strategic choice that avoids fractions and makes the substitution process cleaner. This step is the foundation of the substitution method, allowing us to express one variable as a function of the other.
2. Substitute the expression into the other equation: Next, we substitute the expression for y we found in step 1, which is $y = 4x - 7$, into the second equation, $3y - 12x = -21$. Replacing y with $(4x - 7)$ in the second equation, we get $3(4x - 7) - 12x = -21$. This substitution eliminates y from the second equation, leaving us with an equation in terms of x only. This step is the heart of the substitution method, as it transforms the system of two equations into a single equation that can be solved directly for one variable. The substitution process effectively reduces the complexity of the problem, making it more manageable. By carefully replacing the variable, we ensure that we maintain the equality and can proceed with solving for x.
3. Solve the resulting equation: Now we solve the equation $3(4x - 7) - 12x = -21$ for x. First, distribute the 3 across the terms inside the parentheses: $12x - 21 - 12x = -21$. Next, combine like terms. Notice that $12x$ and $-12x$ cancel each other out, leaving us with $-21 = -21$. This result is a significant one because it indicates that the equation is true for all values of x. This is a characteristic of systems of equations that have infinite solutions. When the variables cancel out and you are left with a true statement, it means that the two equations are essentially the same line, and any point on that line is a solution to the system. This outcome is different from the case where the variables cancel out and you are left with a false statement, which would indicate no solution. Understanding how to interpret these results is crucial in solving systems of equations.
4. Interpret the result: Since we arrived at the identity $-21 = -21$, this means that the two equations are dependent and represent the same line. This indicates that there are infinitely many solutions to the system. Any point that satisfies one equation will also satisfy the other. To visualize this, imagine graphing both equations; you would find that they overlap perfectly. The implication of this is that there is no unique solution that satisfies both equations simultaneously, but rather an infinite number of solutions. This is a key concept in understanding systems of equations and how different outcomes arise when solving them. Recognizing when a system has infinite solutions is just as important as finding a unique solution.

Infinite Solutions Explained

When solving a system of equations, arriving at an identity (a true statement like $-21 = -21$) after substitution indicates that the two equations are dependent. Dependent equations are essentially different forms of the same equation. They represent the same line when graphed, meaning every point on the line is a solution to both equations. This is why we have infinite solutions.

To further illustrate this, let's rewrite the second equation from our system:

$3y - 12x = -21$

Divide the entire equation by 3:

$y - 4x = -7$

Now, add $4x$ to both sides:

$y = 4x - 7$

Notice that this is the same as the expression we found for y in the first equation. This confirms that the two equations are indeed the same line, and any x value will yield a corresponding y value that satisfies both equations. For instance, if we let $x = 0$, then $y = 4(0) - 7 = -7$. So the point $(0, -7)$ is a solution. Similarly, if we let $x = 1$, then $y = 4(1) - 7 = -3$, so the point $(1, -3)$ is also a solution. This process can be repeated indefinitely, generating an infinite number of solutions.

The concept of infinite solutions is a fundamental aspect of linear algebra and is crucial in various applications, including network analysis, circuit design, and linear programming. Recognizing dependent equations is a valuable skill that simplifies problem-solving and provides deeper insights into the relationships between variables. In contrast, if the equations represented parallel lines, there would be no intersection and thus no solution. If they were distinct, intersecting lines, there would be a unique solution. Understanding these scenarios helps in analyzing and solving a wide range of mathematical and real-world problems.

Conclusion

In this article, we explored the substitution method for solving systems of equations. We demonstrated the step-by-step process using the given system:

$egin{array}{c} 4x - y = 7 \ 3y - 12x = -21

\end{array}$

We found that the system has infinite solutions because the two equations are dependent, representing the same line. The key takeaway is that when solving systems of equations, it's essential to recognize the different outcomes: a unique solution, no solution, or infinite solutions. The substitution method is a powerful tool, but understanding the underlying concepts is crucial for correct interpretation and application.

Understanding when a system has infinite solutions is just as important as finding a unique solution. This knowledge allows for a deeper comprehension of the relationships between equations and their graphical representations. It also highlights the importance of careful analysis and interpretation in mathematical problem-solving. The ability to recognize and handle different scenarios makes the substitution method not just a procedural technique but a tool for understanding the nature of systems of equations. By mastering this method, one gains valuable insights into algebraic problem-solving and its applications in various fields.

The substitution method is a fundamental technique in algebra, offering a systematic approach to solving systems of equations. Its application extends beyond the classroom, finding relevance in numerous practical scenarios. Whether it's determining the optimal mix of ingredients in a recipe or analyzing complex financial models, the ability to solve systems of equations is an invaluable skill. By following the step-by-step guide outlined in this article, readers can confidently tackle a wide range of problems, gaining a deeper appreciation for the power and versatility of algebraic methods. The journey through solving systems of equations not only enhances mathematical proficiency but also cultivates critical thinking and problem-solving abilities that are essential in various aspects of life.

Therefore, the correct answer to the given system of equations is D. infinite solutions.