Solving Systems Of Linear Equations Determining Unique, No, Or Infinite Solutions
In linear algebra, understanding the nature of solutions for a system of linear equations is crucial. A system of linear equations can have one solution, no solution, or infinitely many solutions. Determining which case applies involves analyzing the equations and their relationships. This article will explore how to determine the number of solutions for various systems of linear equations.
Methods to Determine the Number of Solutions
There are several methods to determine whether a system of linear equations has one solution, no solution, or infinitely many solutions. The most common methods include:
- Graphical Method: Plotting the equations on a graph and observing the intersection points.
- Substitution Method: Solving one equation for one variable and substituting it into the other equation.
- Elimination Method: Adding or subtracting multiples of the equations to eliminate one variable.
- Determinant Method: Using determinants of matrices formed by the coefficients of the variables.
1. Graphical Method
The graphical method involves plotting the lines represented by the linear equations on a coordinate plane. The number of intersection points indicates the number of solutions:
- One Solution: The lines intersect at a single point.
- No Solution: The lines are parallel and do not intersect.
- Infinitely Many Solutions: The lines are coincident (overlap each other).
While the graphical method provides a visual understanding, it may not always be precise, especially when the solutions are not integers.
2. Substitution Method
The substitution method involves solving one of the equations for one variable and substituting that expression into the other equation. This results in a single equation with one variable, which can be solved. The value of this variable is then substituted back into one of the original equations to find the value of the other variable. The process helps to reveal the nature of solutions:
- One Solution: A unique solution for both variables is found.
- No Solution: The substitution leads to a contradiction (e.g., 0 = 1).
- Infinitely Many Solutions: The substitution leads to an identity (e.g., 0 = 0).
To elaborate on the substitution method, consider a system of equations:
ax + by = c
dx + ey = f
To apply the substitution method effectively, you would first choose one equation and solve for one variable in terms of the other. For example, let's say we pick the first equation and solve for x:
ax = c - by
x = (c - by) / a
Now, substitute this expression for x into the second equation:
d((c - by) / a) + ey = f
This equation now contains only one variable, y, and can be solved to find the value of y. Once you have the value of y, you can substitute it back into either of the original equations or the expression you derived for x to find the value of x. For instance, using the expression x = (c - by) / a, you can plug in the value of y to calculate x:
x = (c - b*y_value) / a
If this process yields unique values for both x and y, the system has one solution. However, if at any point during the substitution and simplification process you encounter a contradiction, such as 0 = 1, it indicates that the system has no solution, as there are no values of x and y that can satisfy both equations simultaneously. On the other hand, if the substitution leads to an identity, like 0 = 0, it means that the two equations are dependent and represent the same line, resulting in infinitely many solutions.
3. Elimination Method
The elimination method involves manipulating the equations so that, when added or subtracted, one of the variables is eliminated. This can be achieved by multiplying one or both equations by a constant to make the coefficients of one variable the same or opposites. Similar to the substitution method, this method reveals the nature of the solutions:
- One Solution: A unique solution for both variables is found.
- No Solution: The elimination leads to a contradiction (e.g., 0 = 1).
- Infinitely Many Solutions: The elimination leads to an identity (e.g., 0 = 0).
To further illustrate the elimination method, let's consider a system of linear equations:
2x + 3y = 10
4x - y = 2
The goal of the elimination method is to manipulate the equations in such a way that, when they are added or subtracted, one of the variables cancels out. This simplifies the system, allowing us to solve for the remaining variable. In this specific example, we can eliminate the y variable. To do this, we first need to make the coefficients of y in both equations the same magnitude but with opposite signs. We can multiply the second equation by 3:
3 * (4x - y) = 3 * 2
12x - 3y = 6
Now, we have the modified system:
2x + 3y = 10
12x - 3y = 6
Next, we add the two equations:
(2x + 3y) + (12x - 3y) = 10 + 6
This simplifies to:
14x = 16
Now, we can solve for x:
x = 16 / 14
x = 8 / 7
With the value of x determined, we can substitute it back into one of the original equations to solve for y. Let's use the first original equation:
2x + 3y = 10
2 * (8 / 7) + 3y = 10
16 / 7 + 3y = 10
To isolate y, subtract 16/7 from both sides:
3y = 10 - 16 / 7
3y = (70 - 16) / 7
3y = 54 / 7
Now, divide by 3:
y = (54 / 7) / 3
y = 18 / 7
Thus, we have found the unique values for both x and y: x = 8/7 and y = 18/7. This indicates that the system has one unique solution. The process of eliminating variables and solving for the remaining ones demonstrates the core principle of the elimination method. If, instead of finding unique values, we encountered a contradiction (e.g., 0 = 1) or an identity (e.g., 0 = 0), we would conclude that the system has no solution or infinitely many solutions, respectively.
4. Determinant Method
The determinant method uses determinants of matrices to determine the number of solutions. For a system of two linear equations:
ax + by = c
dx + ey = f
We can form a coefficient matrix:
A = | a b |
| d e |
The determinant of A (denoted as |A|) is calculated as:
|A| = ae - bd
Based on the determinant, the nature of solutions can be determined:
- One Solution: If |A| ≠0.
- No Solution or Infinitely Many Solutions: If |A| = 0, further analysis is needed.
If |A| = 0, we compute determinants of matrices formed by replacing the columns of A with the constant terms:
Ax = | c b |
| f e |
Ay = | a c |
| d f |
If |Ax| = |Ay| = 0, the system has infinitely many solutions. If either |Ax| ≠0 or |Ay| ≠0, the system has no solution.
To further clarify the determinant method, let's consider a system of linear equations:
2x + 3y = 7
4x + 6y = 14
First, we form the coefficient matrix A using the coefficients of x and y:
A = | 2 3 |
| 4 6 |
The determinant of A, denoted as |A|, is calculated as follows:
|A| = (2 * 6) - (3 * 4)
|A| = 12 - 12
|A| = 0
Since |A| = 0, this indicates that the system either has no solution or infinitely many solutions. To determine which case applies, we need to compute the determinants of the matrices formed by replacing the columns of A with the constant terms from the original equations. We define Ax as the matrix formed by replacing the first column of A (the coefficients of x) with the constants, and Ay as the matrix formed by replacing the second column of A (the coefficients of y) with the constants. Thus:
Ax = | 7 3 |
| 14 6 |
Ay = | 2 7 |
| 4 14 |
Now, we calculate the determinants |Ax| and |Ay|:
|Ax| = (7 * 6) - (3 * 14)
|Ax| = 42 - 42
|Ax| = 0
|Ay| = (2 * 14) - (7 * 4)
|Ay| = 28 - 28
|Ay| = 0
Since both |Ax| and |Ay| are equal to 0, this signifies that the system has infinitely many solutions. The zero determinants indicate that the two equations are linearly dependent, meaning one equation is a multiple of the other, and they represent the same line on a graph. This graphical representation results in the lines overlapping, hence infinitely many points of intersection, each representing a solution to the system.
Examples and Solutions
Let's apply these methods to the given systems of equations.
Example 1
3x - 2y = 3
6x - 4y = 1
Using the determinant method:
A = | 3 -2 |
| 6 -4 |
|A| = (3 * -4) - (-2 * 6) = -12 + 12 = 0
Ax = | 3 -2 |
| 1 -4 |
|Ax| = (3 * -4) - (-2 * 1) = -12 + 2 = -10
Since |A| = 0 and |Ax| ≠0, the system has no solution.
Example 2
3x - 5y = 8
5x - 3y = 2
Using the determinant method:
A = | 3 -5 |
| 5 -3 |
|A| = (3 * -3) - (-5 * 5) = -9 + 25 = 16
Since |A| ≠0, the system has one solution.
Example 3
3x + 2y = 8
4x + 3y = 1
Using the determinant method:
A = | 3 2 |
| 4 3 |
|A| = (3 * 3) - (2 * 4) = 9 - 8 = 1
Since |A| ≠0, the system has one solution.
Example 4
3x - 6y = 3
2x - 4y = 2
Using the determinant method:
A = | 3 -6 |
| 2 -4 |
|A| = (3 * -4) - (-6 * 2) = -12 + 12 = 0
Ax = | 3 -6 |
| 2 -4 |
|Ax| = (3 * -4) - (-6 * 2) = -12 + 12 = 0
Ay = | 3 3 |
| 2 2 |
|Ay| = (3 * 2) - (3 * 2) = 6 - 6 = 0
Since |A| = |Ax| = |Ay| = 0, the system has infinitely many solutions.
Example 5
3x - 4y = 2
6x - 8y = 4
Using the determinant method:
A = | 3 -4 |
| 6 -8 |
|A| = (3 * -8) - (-4 * 6) = -24 + 24 = 0
Ax = | 2 -4 |
| 4 -8 |
|Ax| = (2 * -8) - (-4 * 4) = -16 + 16 = 0
Ay = | 3 2 |
| 6 4 |
|Ay| = (3 * 4) - (2 * 6) = 12 - 12 = 0
Since |A| = |Ax| = |Ay| = 0, the system has infinitely many solutions.
Conclusion
Determining whether a system of linear equations has one solution, no solution, or infinitely many solutions is a fundamental concept in linear algebra. The graphical, substitution, elimination, and determinant methods provide different approaches to analyzing the relationships between equations. Understanding these methods allows for effective problem-solving and a deeper comprehension of linear systems.
By systematically applying these techniques, one can accurately determine the nature of solutions for any given system of linear equations, whether in academic exercises or real-world applications.
