Analyzing And Solving Systems Of Equations System A System B And System C
Understanding systems of equations is a fundamental concept in algebra, crucial for solving various mathematical and real-world problems. In this article, we will delve into the analysis of three different systems of equations – System A, System B, and System C – to determine the truthfulness of given statements. We will employ methods such as substitution, elimination, and graphical analysis to dissect these systems and arrive at accurate conclusions. Mastering the art of solving systems of equations not only strengthens one's algebraic skills but also lays a solid foundation for more advanced mathematical concepts. So, let's embark on this journey of algebraic exploration and uncover the solutions hidden within these equations.
System A: Solving 2x - 3y = 4 and 3x - 4y = 5
To begin our analysis, let's focus on System A, which comprises the equations 2x - 3y = 4 and 3x - 4y = 5. Our goal is to find the values of x and y that satisfy both equations simultaneously. We can employ several methods to solve this system, such as substitution or elimination. For this demonstration, let's use the elimination method. The elimination method involves manipulating the equations to eliminate one variable, allowing us to solve for the other. To eliminate y, we can multiply the first equation by 4 and the second equation by 3. This gives us:
(2x - 3y = 4) * 4 => 8x - 12y = 16 (3x - 4y = 5) * 3 => 9x - 12y = 15
Now, we subtract the second modified equation from the first:
(8x - 12y) - (9x - 12y) = 16 - 15
Simplifying this, we get:
-x = 1
Therefore, x = -1. Now that we have the value of x, we can substitute it back into either of the original equations to find y. Let's use the first equation, 2x - 3y = 4:
2(-1) - 3y = 4 -2 - 3y = 4 -3y = 6 y = -2
Thus, the solution to System A is x = -1 and y = -2. This unique solution indicates that the lines represented by these equations intersect at a single point on the coordinate plane. The elimination method proved effective in systematically isolating and determining the values of x and y. Understanding this process is crucial for tackling more complex systems of equations. The solution (x = -1, y = -2) is a critical point that satisfies both equations in System A, making it the definitive solution for this system.
System B: Solving 2x - 3y = 4 and 4x - y = 18
Next, we turn our attention to System B, which consists of the equations 2x - 3y = 4 and 4x - y = 18. Similar to System A, we aim to find the values of x and y that satisfy both equations concurrently. Again, we can use either the substitution or elimination method. For System B, let's demonstrate the substitution method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with one variable. From the second equation, 4x - y = 18, we can solve for y:
y = 4x - 18
Now, substitute this expression for y into the first equation, 2x - 3y = 4:
2x - 3(4x - 18) = 4
Expanding and simplifying, we get:
2x - 12x + 54 = 4 -10x = -50 x = 5
Now that we have the value of x, we can substitute it back into the expression we found for y:
y = 4(5) - 18 y = 20 - 18 y = 2
Therefore, the solution to System B is x = 5 and y = 2. This distinct solution, different from that of System A, suggests that the lines represented by the equations in System B intersect at a different point on the coordinate plane. The substitution method provided a straightforward approach to solving this system, highlighting the flexibility and versatility of algebraic techniques. This solution (x = 5, y = 2) is the unique point of intersection for the lines defined by the equations in System B.
System C: Analyzing y = 5x + 3 and 12x - 3y = 54
Finally, let's investigate System C, defined by the equations y = 5x + 3 and 12x - 3y = 54. This system presents an interesting scenario that warrants careful analysis. We can start by simplifying the second equation to see if it reveals any insights. Divide the second equation, 12x - 3y = 54, by 3:
4x - y = 18
Now, let's compare this simplified equation with the first equation, y = 5x + 3. We can use the substitution method here as well. Substitute the expression for y from the first equation into the simplified second equation:
4x - (5x + 3) = 18
Simplifying, we get:
4x - 5x - 3 = 18 -x = 21 x = -21
Now, substitute the value of x back into the equation y = 5x + 3:
y = 5(-21) + 3 y = -105 + 3 y = -102
Thus, the solution to System C is x = -21 and y = -102. This unique solution indicates that the two lines represented by the equations in System C intersect at a single point. System C's solution is particularly interesting as it demonstrates how simplification and substitution can be used to solve seemingly complex systems. The solution (x = -21, y = -102) provides a definitive intersection point for the lines represented by the equations in System C.
Comparing the Solutions and Analyzing the Statements
Now that we have solved each system, let's compare the solutions and analyze the statements provided. We found the following solutions:
- System A: x = -1, y = -2
- System B: x = 5, y = 2
- System C: x = -21, y = -102
The first statement, "Systems A and B have different solutions," is true. System A has the solution (-1, -2), while System B has the solution (5, 2). These solutions are clearly distinct. The difference in solutions highlights the unique nature of each system and the specific intersection points of their corresponding lines.
The second statement, "System C simplifies to 2x - 3y = 4," requires further examination. We started with the equations y = 5x + 3 and 12x - 3y = 54. We simplified the second equation to 4x - y = 18. The first equation given in System C is y = 5x + 3. To ascertain if it simplifies to 2x - 3y, we would have to have that equation be the same equation. So lets's multiply the given equation, y = 5x + 3, and see what we get: 2x - 3(5x+3) = 4. Simplify: 2x - 15x - 9 = 4, -13x = 13, x = -1. Now we know the second equation is 4x - y = 18, so 4(-1) - y = 18, -4 - y = 18, y = -22, but that's not the given equation, so this system does not simplify to 2x - 3y = 4, making this statement false.
Conclusion
In conclusion, by systematically solving and analyzing the three systems of equations, we have determined the truthfulness of the given statements. We used methods like elimination and substitution to find the unique solutions for each system. Our analysis confirms that Systems A and B indeed have different solutions, while System C does not simplify to the equation 2x - 3y = 4. This exercise underscores the importance of algebraic techniques in solving systems of equations and provides valuable insights into the nature of linear equations and their solutions. Understanding these concepts is crucial for success in algebra and related fields.
