Solving Systems Of Equations A Comprehensive Guide
When faced with a system of equations, the task is to find the values of the variables that satisfy all equations simultaneously. This article delves into solving the provided system of equations, explaining each step in detail to ensure a clear understanding of the process. We will explore how adding the equations leads to a simplified form and ultimately helps in finding the solution. Let's embark on this mathematical journey together!
Understanding the System of Equations
The system of equations presented is:
$ \begin{array}{l} 2. 5 y+3 x=27 \ 3. x-2.5 y=5 \end{array} $
This system comprises two linear equations with two variables, x and y. Our goal is to find the values of x and y that make both equations true. There are several methods to solve such systems, including substitution, elimination, and graphing. In this case, we will focus on the elimination method, which involves adding or subtracting the equations to eliminate one variable.
To effectively use the elimination method, we aim to manipulate the equations so that the coefficients of one variable are opposites. This way, when we add the equations, that variable will be eliminated. Looking at our system, we notice that the coefficients of y are 2.5 and -2.5, which are already opposites. This makes the elimination method particularly straightforward in this scenario.
Before proceeding with the addition, it's beneficial to rewrite the equations in a standard form (Ax + By = C) to maintain clarity and organization. This helps in visualizing the coefficients and constants, reducing the chances of making errors during the process. Let’s rewrite the equations:
$ \begin{array}{l} 3. x + 2.5 y = 27 \ 4. x - 2.5 y = 5 \end{array} $
Now, the equations are neatly aligned, and we can clearly see the coefficients of x and y. The next step involves adding these equations together, which will eliminate the y variable, allowing us to solve for x. Understanding this setup is crucial for the subsequent steps, as it lays the foundation for finding the solution to the system.
Adding the Equations: A Step-by-Step Guide
Now, let’s add the two equations together. This is a crucial step in the elimination method, as it allows us to get rid of one variable and simplify the problem. We start with the equations in their standard form:
$ \begin{array}{l} 3 x + 2.5 y = 27 \ 5 x - 2.5 y = 5 \end{array} $
When adding equations, we add the corresponding sides separately. That means we add the left-hand sides (LHS) together and the right-hand sides (RHS) together. This ensures that the equality is maintained throughout the process. Adding the LHS, we have (3x + 2.5y) + (5x - 2.5y). On the RHS, we have 27 + 5.
Let's perform the addition step by step:
- Combine the x terms: 3x + 5x = 8x
- Combine the y terms: 2.5y + (-2.5y) = 0
- Combine the constants: 27 + 5 = 32
Notice that the y terms cancel each other out, which is exactly what we intended. This is the power of the elimination method – by carefully adding or subtracting equations, we can eliminate variables and make the system easier to solve. After adding, we are left with a single equation in one variable:
8x = 32
This equation is much simpler than the original system. Now, we can easily solve for x by dividing both sides of the equation by 8:
x = 32 / 8 x = 4
So, we have found the value of x. This is a significant milestone in solving the system. However, we are not done yet. We still need to find the value of y. To do this, we can substitute the value of x into one of the original equations and solve for y. This substitution process is the next key step in completing our solution.
Solving for x and the Resulting Equation
As demonstrated in the previous section, adding the two equations eliminates the y variable, resulting in a simpler equation that allows us to solve for x. Let’s reiterate the process to reinforce understanding.
Starting with the system:
$ \begin{array}{l} 3 x + 2.5 y = 27 \ 5 x - 2.5 y = 5 \end{array} $
Adding the equations, we get:
(3x + 2.5y) + (5x - 2.5y) = 27 + 5
Combining like terms:
8x = 32
Now, we solve for x by dividing both sides by 8:
x = 32 / 8 x = 4
Therefore, the value of x is 4. The equation resulting from adding the two original equations is:
8x = 32
This equation is a direct consequence of adding the two original equations and eliminating y. It provides a straightforward way to find the value of x. Now that we have found x, the next step is to use this value to find y. We can do this by substituting x = 4 into either of the original equations. Let’s choose the first equation for this purpose:
3x + 2.5y = 27
Substitute x = 4:
3(4) + 2.5y = 27 12 + 2.5y = 27
Now, we need to isolate y. Subtract 12 from both sides:
- 5y = 27 - 12
- 5y = 15
Finally, divide both sides by 2.5 to solve for y:
y = 15 / 2.5 y = 6
So, the value of y is 6. We have now found both x and y. The solution to the system of equations is x = 4 and y = 6. It's essential to verify this solution by substituting these values back into both original equations to ensure they hold true. This verification step confirms the accuracy of our solution.
Finding the Solution to the System
Having determined the value of x to be 4, we now proceed to find the value of y. As discussed earlier, we substitute x = 4 into one of the original equations. Let’s use the first equation:
Substitute x = 4:
This simplifies to:
To isolate the term with y, we subtract 12 from both sides of the equation:
Which gives us:
Now, to solve for y, we divide both sides by 2.5:
Performing the division, we find:
Thus, the value of y is 6. Now that we have both x = 4 and y = 6, we can express the solution to the system of equations as an ordered pair (x, y), which is (4, 6). However, it is crucial to verify this solution to ensure its accuracy.
To verify, we substitute x = 4 and y = 6 into both original equations:
For the first equation:
The first equation holds true.
For the second equation:
The second equation also holds true. Since the solution (4, 6) satisfies both equations, we can confidently conclude that it is the correct solution to the system. This comprehensive step-by-step approach ensures clarity and accuracy in solving systems of equations.
Conclusion: The Solution and Its Significance
In conclusion, we have successfully solved the given system of equations:
$ \begin{array}{l} 3. 5 y+3 x=27 \ 4. x-2.5 y=5 \end{array} $
By employing the elimination method, we added the two equations together, which resulted in the elimination of the y variable. This led us to the equation:
8x = 32
Solving for x, we found:
x = 4
Subsequently, we substituted x = 4 into one of the original equations to solve for y, which gave us:
y = 6
Therefore, the solution to the system of equations is the ordered pair (4, 6). This means that the values x = 4 and y = 6 satisfy both equations simultaneously. We verified this solution by substituting these values back into the original equations, confirming their validity.
The significance of solving systems of equations extends beyond mere mathematical exercises. These systems often arise in real-world applications, such as in physics, engineering, economics, and computer science. For instance, they can be used to model supply and demand curves in economics, determine the forces acting on a structure in engineering, or solve optimization problems in computer science.
Understanding how to solve systems of equations is a fundamental skill in mathematics and is essential for tackling more complex problems in various fields. The elimination method, as demonstrated in this article, is a powerful tool for solving such systems, especially when the coefficients of one variable are opposites or can be easily made opposites through multiplication. By mastering these techniques, one can confidently approach and solve a wide range of mathematical and real-world problems. This step-by-step guide provides a solid foundation for further exploration and application of these concepts.
