System Of Equations: Did Julio Find The Correct Solution?
When dealing with systems of equations, it's crucial to verify if a given point is indeed a solution. In this article, we'll delve into a scenario where Julio tests the point (4,-2) to see if it satisfies the following system of equations:
We'll analyze Julio's work step-by-step to determine if he arrived at the correct conclusion. This involves substituting the x and y values of the point into both equations and checking if they hold true. Understanding how to verify solutions is fundamental in algebra, and this example will provide a clear illustration of the process. So, let's get started and examine Julio's approach to solving this problem.
Understanding Systems of Equations and Solutions
Before we dive into Julio's work, let's clarify what a system of equations is and what it means for a point to be a solution. A system of equations is a set of two or more equations containing the same variables. A solution to a system of equations is a set of values for the variables that makes all the equations in the system true simultaneously. Graphically, a solution represents the point where the lines or curves of the equations intersect.
To check if a point is a solution, we substitute the coordinates of the point into each equation in the system. If the point satisfies all equations, then it is a solution to the system. If it fails to satisfy even one equation, it is not a solution.
In our case, we have two linear equations:
- -3x - 2y = -8
- y = 2x - 5
Julio tested the point (4,-2). This means he substituted x = 4 and y = -2 into both equations to see if they hold true. Let's examine his work closely to identify any potential errors.
Analyzing Julio's Work
Julio's work for the first equation is as follows:
So far, Julio's work looks correct for the first equation. He correctly substituted x = 4 and y = -2 into the equation -3x - 2y = -8 and simplified the expression. The result, -8 = -8, is a true statement, indicating that the point (4,-2) satisfies the first equation. This is a critical step in verifying the solution, as the point must satisfy every equation in the system to be considered a valid solution. It's always a good practice to double-check each step to avoid any arithmetic errors that could lead to an incorrect conclusion. Now, let's move on to the second equation to see if the point satisfies it as well.
However, Julio only tested the point in the first equation. To determine if is a solution to the system of equations, he needs to test it in both equations. This is a common mistake when solving systems of equations, where individuals might stop after verifying the solution in one equation. The point must satisfy all equations in the system to be a valid solution.
Let's test the point (4, -2) in the second equation, y = 2x - 5, to complete the solution verification.
Testing the Point in the Second Equation
The second equation in the system is:
To test if the point (4, -2) is a solution, we substitute x = 4 and y = -2 into this equation:
The result, -2 = 3, is a false statement. This means that the point (4, -2) does not satisfy the second equation. Since the point does not satisfy both equations in the system, it is not a solution to the system of equations.
This step is crucial in understanding why Julio's conclusion was incorrect. While the point (4, -2) satisfied the first equation, it failed to satisfy the second. For a point to be a valid solution to a system of equations, it must hold true for all equations within that system. By identifying this discrepancy, we can confidently say that Julio's initial assessment was flawed due to incomplete verification.
Conclusion: Did Julio Make a Mistake?
Yes, Julio made a mistake. While he correctly verified that the point (4, -2) is a solution to the first equation, he failed to check the second equation. For a point to be a solution to a system of equations, it must satisfy all equations in the system. Since (4, -2) does not satisfy the equation y = 2x - 5, it is not a solution to the system.
This example highlights the importance of thoroughness when solving mathematical problems. It’s essential to verify solutions in all equations of a system to ensure accuracy. Overlooking even one equation can lead to incorrect conclusions. By carefully checking each step and ensuring that all conditions are met, we can confidently arrive at the correct solution. This attention to detail is a key aspect of mathematical problem-solving and critical thinking.
In conclusion, Julio incorrectly determined that (4, -2) is a solution to the system of equations because he only tested the point in one equation. To correctly verify solutions to systems of equations, remember to substitute the point into every equation in the system. For further learning on systems of equations, you can visit Khan Academy's Systems of Equations Section.
