Determine Whether The Point (0, 3) Is A Solution To The System Of Equations X - 3y = 3 And 2x - 9y = 3.
To determine whether the points (0, 3), (4, -3), and (9, 6) are solutions to the given system of equations, we need to substitute the x and y coordinates of each point into both equations and check if the equations hold true. The system of equations is:
- x - 3y = 3
- 2x - 9y = 3
Let’s analyze each point individually.
Point (0, 3)
To check if the point (0, 3) is a solution, we substitute x = 0 and y = 3 into both equations.
Equation 1: x - 3y = 3
Substituting the values, we get:
0 - 3(3) = 3
0 - 9 = 3
-9 = 3
This statement is false. Therefore, the point (0, 3) does not satisfy the first equation.
Equation 2: 2x - 9y = 3
Substituting the values, we get:
2(0) - 9(3) = 3
0 - 27 = 3
-27 = 3
This statement is also false. Since the point (0, 3) does not satisfy either equation, it is not a solution to the system of equations. In summary, the point (0,3) fails to meet the criteria for both equations within the system. When x is 0 and y is 3, the first equation x - 3y = 3 results in -9 = 3, which is a false statement. Similarly, substituting these values into the second equation 2x - 9y = 3 gives -27 = 3, also a false statement. For a point to be considered a solution to a system of equations, it must satisfy all equations in that system. Since (0,3) satisfies neither, it's definitively not a solution. This illustrates a foundational concept in solving systems of equations: each equation represents a condition that must be simultaneously met for a point to be considered a valid solution. The failure of (0,3) to meet these conditions underscores the importance of methodical substitution and verification in mathematical problem-solving.
Point (4, -3)
Next, we check if the point (4, -3) is a solution by substituting x = 4 and y = -3 into both equations.
Equation 1: x - 3y = 3
Substituting the values, we get:
4 - 3(-3) = 3
4 + 9 = 3
13 = 3
This statement is false. Thus, the point (4, -3) does not satisfy the first equation.
Equation 2: 2x - 9y = 3
Substituting the values, we get:
2(4) - 9(-3) = 3
8 + 27 = 3
35 = 3
This statement is also false. Since the point (4, -3) does not satisfy either equation, it is not a solution to the system of equations. Analyzing the point (4, -3) further emphasizes the stringent requirements for a solution to a system of equations. Substituting x = 4 and y = -3 into the first equation, x - 3y = 3, we arrive at 13 = 3, which is clearly not true. Similarly, substituting these values into the second equation, 2x - 9y = 3, results in 35 = 3, another false statement. The consistency in the failure of this point across both equations reinforces the idea that a valid solution must simultaneously satisfy all equations within the system. This process of substitution and verification is crucial in linear algebra, where the solutions to systems of equations represent intersections of lines (in two dimensions) or planes (in three dimensions), and such intersections must inherently lie on all the lines or planes involved. The methodical approach demonstrated here is fundamental for accurately identifying such intersections.
Point (9, 6)
Now, let's check if the point (9, 6) is a solution by substituting x = 9 and y = 6 into both equations.
Equation 1: x - 3y = 3
Substituting the values, we get:
9 - 3(6) = 3
9 - 18 = 3
-9 = 3
This statement is false. Therefore, the point (9, 6) does not satisfy the first equation.
Equation 2: 2x - 9y = 3
Substituting the values, we get:
2(9) - 9(6) = 3
18 - 54 = 3
-36 = 3
This statement is also false. Since the point (9, 6) does not satisfy either equation, it is not a solution to the system of equations. Evaluating the point (9, 6) through substitution provides further insight into the nature of solutions for systems of equations. Inserting x = 9 and y = 6 into the first equation, x - 3y = 3, leads to -9 = 3, a clear contradiction. The second equation, 2x - 9y = 3, upon substitution, yields -36 = 3, another false statement. The consistent failure of this point to satisfy both equations underscores a critical aspect of linear systems: a solution must be a point that, when its coordinates are substituted into the equations, makes all the equations true simultaneously. In this case, the point (9, 6) fails to meet this criterion for either equation, thereby confirming it is not a solution to the system. This methodical verification process is a cornerstone of algebraic problem-solving, ensuring accuracy and a deep understanding of the conditions under which a set of equations holds true.
Conclusion
In conclusion, none of the points (0, 3), (4, -3), and (9, 6) are solutions to the given system of equations because none of them satisfy both equations simultaneously. To solve this system of equations, one might use methods such as substitution, elimination, or matrix operations to find the actual solution(s) if they exist. In summary, determining whether a point is a solution to a system of equations requires methodical substitution and verification. None of the tested points, (0, 3), (4, -3), and (9, 6), met the necessary criteria of satisfying both equations simultaneously. This process highlights the importance of precision in algebraic problem-solving and the need for a comprehensive understanding of how systems of equations work.