Solve The System Of Equations: X - Y = 0, X + 2y = 9. What Type Of System Is It (no Solution, One Solution, Or Infinite Solutions)?

Jun 19, 2025 by ADMIN 132 views

Solving a System of Equations: A Step-by-Step Guide

In the realm of mathematics, systems of equations play a crucial role in modeling real-world scenarios and finding solutions that satisfy multiple conditions simultaneously. This article delves into the process of solving a specific system of equations, providing a comprehensive explanation and step-by-step guidance. We'll explore the fundamental concepts, techniques, and interpretations involved in solving systems of equations, ensuring a clear understanding for readers of all backgrounds. Understanding the underlying principles is paramount for tackling more complex mathematical problems in various fields.

The system of equations we will be tackling is:

x - y = 0
x + 2y = 9

We will solve this system using two common methods: substitution and elimination. Each method offers a unique approach to finding the solution, providing flexibility and alternative strategies for tackling different systems of equations. Our primary objective is not just to find the numerical solution but also to illustrate the underlying concepts and techniques that make these methods effective. By exploring both methods, we aim to equip you with a versatile toolkit for solving a wide array of systems of equations.

Before we dive into the solution, it's essential to identify the type of system we're dealing with. Systems of equations can fall into three categories: those with no solutions, those with exactly one solution, and those with infinite solutions. Determining the type of system beforehand can provide valuable insights into the nature of the problem and guide our solution process. By understanding the characteristics of each type of system, we can avoid unnecessary computations and focus on the most efficient approach.

Identifying the Type of System

To identify the type of system, we can analyze the equations graphically or algebraically. Graphically, each equation represents a line, and the solution to the system corresponds to the point(s) where the lines intersect. If the lines are parallel, there is no solution. If the lines intersect at a single point, there is exactly one solution. If the lines coincide, there are infinite solutions. Algebraically, we can analyze the coefficients of the variables. If the equations are inconsistent (i.e., lead to a contradiction), there is no solution. If the equations are independent and consistent, there is exactly one solution. If the equations are dependent, there are infinite solutions.

In our case, we can rewrite the first equation as x = y. Substituting this into the second equation, we get y + 2y = 9, which simplifies to 3y = 9. This equation has a unique solution for y, which implies that the system has exactly one solution. This preliminary analysis not only confirms the existence of a solution but also provides a roadmap for our next steps. By establishing the type of system upfront, we can streamline our solution process and avoid unnecessary detours.

Method 1: Substitution

The substitution method involves 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 using standard algebraic techniques. The substitution method is particularly effective when one of the equations can be easily solved for one variable in terms of the other.

Solve the first equation for x: From the equation x - y = 0, we can easily isolate x by adding y to both sides:
```
x = y
```
This simple step is the foundation of the substitution method, allowing us to express one variable in terms of the other. The ability to isolate variables is a fundamental skill in algebra and is essential for solving various types of equations.
Substitute this expression for x into the second equation: Now, we substitute x = y into the second equation, x + 2y = 9:
```
y + 2y = 9
```
This substitution is a crucial step in reducing the system to a single equation with one variable. By replacing x with its equivalent expression in terms of y, we eliminate one variable and pave the way for solving for the remaining variable.
Solve for y: Combine like terms:
```
3y = 9
```
Divide both sides by 3:
```
y = 3
```
This step provides us with the value of y, which is one part of the solution to the system. The ability to solve simple algebraic equations is a fundamental skill in mathematics and is essential for tackling more complex problems.
Substitute the value of y back into the equation x = y to find x: Since x = y and y = 3, we have:
```
x = 3
```
This final substitution gives us the value of x, completing the solution to the system. The simplicity of this step underscores the power of the substitution method in efficiently finding the solution to a system of equations.

Therefore, the solution to the system of equations using the substitution method is x = 3 and y = 3. This means that the point (3, 3) is the intersection of the two lines represented by the equations in the system. The substitution method, as demonstrated here, offers a systematic approach to solving systems of equations by reducing the problem to a single equation with one variable.

Method 2: Elimination

The elimination method, also known as the addition method, involves manipulating the equations so that the coefficients of one variable are opposites. By adding the equations together, one variable is eliminated, leaving a single equation with one variable. This method is particularly useful when the coefficients of one variable are already opposites or can be easily made opposites through multiplication.

Observe the equations:
```
x - y = 0
x + 2y = 9
```
Notice that the coefficients of x are already the same (both are 1). This makes the elimination method particularly convenient in this case. Recognizing patterns and identifying opportunities for simplification is a crucial skill in mathematics.
Multiply the first equation by -1: This will make the coefficients of x opposites:
```
-1(x - y) = -1(0)
-x + y = 0
```
Multiplying an equation by a constant is a fundamental algebraic operation that preserves the equality. In this case, multiplying by -1 allows us to create opposite coefficients for the x variable, which is the key to the elimination method.
Add the modified first equation to the second equation:
```
(-x + y) + (x + 2y) = 0 + 9
```
This step eliminates the x variable, leaving us with an equation in terms of y only.
Simplify and solve for y:
```
3y = 9
y = 3
```
This step provides us with the value of y, which is one part of the solution to the system. The simplicity of this step highlights the effectiveness of the elimination method in reducing a system of equations to a single equation with one variable.
Substitute the value of y back into either of the original equations to solve for x: Using the first equation, x - y = 0:
```
x - 3 = 0
x = 3
```
This final substitution gives us the value of x, completing the solution to the system. The ability to substitute known values into equations is a fundamental skill in algebra and is essential for solving various types of problems.

Therefore, the solution to the system of equations using the elimination method is x = 3 and y = 3. This confirms the solution we obtained using the substitution method, reinforcing the accuracy and consistency of both methods. The elimination method, as demonstrated here, offers an alternative approach to solving systems of equations by strategically eliminating variables to simplify the problem.

Conclusion

In conclusion, the system of equations

x - y = 0
x + 2y = 9

has exactly one solution, which is x = 3 and y = 3. We arrived at this solution using both the substitution and elimination methods, demonstrating the versatility and power of these techniques. Understanding these methods is essential for solving a wide range of mathematical problems and for modeling real-world scenarios. The ability to solve systems of equations is a fundamental skill in mathematics and has applications in various fields, including science, engineering, economics, and computer science. By mastering these techniques, you gain a valuable tool for tackling complex problems and making informed decisions.

This article has provided a step-by-step guide to solving a system of equations, emphasizing the importance of understanding the underlying concepts and techniques. We explored two common methods, substitution and elimination, and demonstrated their application to a specific system of equations. By understanding these methods and their variations, you can confidently tackle a wide range of systems of equations and apply these skills to real-world problems. Remember, practice is key to mastering these techniques, so be sure to work through various examples and challenge yourself to solve increasingly complex systems of equations. The journey of mathematical discovery is ongoing, and the ability to solve systems of equations is a valuable step along the way.