To determine how many solutions a linear system has, you need to analyze the relationships between the equations within the system. There are three possibilities: no solution, one solution, or infinitely many solutions.
Understanding the Possibilities
Let's consider a system of linear equations, typically represented graphically as lines. The number of solutions corresponds to the number of intersection points between these lines.
- No Solution: The lines are parallel and do not intersect. The system is inconsistent.
- One Solution: The lines intersect at exactly one point. The system is consistent and independent.
- Infinitely Many Solutions: The lines are the same (coincident). Every point on the line is a solution. The system is consistent and dependent.
Methods to Determine the Number of Solutions
Here's how you can determine the number of solutions, depending on how the linear system is presented:
1. Graphical Method
- Graph the equations: Plot each equation on a coordinate plane.
- Observe the intersection:
- If the lines are parallel, there are no solutions.
- If the lines intersect at one point, there is one solution.
- If the lines are the same line, there are infinitely many solutions.
2. Algebraic Methods
a. Substitution or Elimination
- Solve for a variable: Use substitution or elimination to try and solve for the variables in the system.
- Analyze the result:
- No Solution: If you arrive at a contradiction (e.g., 0 = 1), the system has no solution.
- One Solution: If you can solve for unique values for each variable, the system has one solution.
- Infinitely Many Solutions: If you arrive at an identity (e.g., 0 = 0) and can express one variable in terms of the other, the system has infinitely many solutions.
b. Determinants (for systems with the same number of equations as variables)
- Form the coefficient matrix: Create a matrix using the coefficients of the variables in the equations.
- Calculate the determinant: Find the determinant of the coefficient matrix.
- Non-zero determinant: The system has exactly one solution.
- Zero determinant: The system has either no solution or infinitely many solutions. Further investigation (e.g., using substitution or elimination) is needed to distinguish between these two cases. You could calculate other determinants to see if the system is consistent.
3. Row Echelon Form (REF) or Reduced Row Echelon Form (RREF)
- Write the augmented matrix: Represent the system as an augmented matrix.
- Perform row operations: Use Gaussian elimination or Gauss-Jordan elimination to transform the matrix into REF or RREF.
- Analyze the result:
- No Solution: If you have a row in the form [0 0 ... 0 | b] where b is non-zero, the system has no solution.
- One Solution: If you have a pivot (leading 1) in every column corresponding to a variable, the system has one solution.
- Infinitely Many Solutions: If you have at least one column corresponding to a variable without a pivot, the system has infinitely many solutions. These variables are "free variables," meaning their values can be chosen arbitrarily.
Example:
Consider the system:
- x + y = 2
- 2x + 2y = 4
- Graphical Method: If you graph these, you'll find they are the same line.
- Algebraic Method (Substitution): Solve equation 1 for x: x = 2 - y. Substitute into equation 2: 2(2-y) + 2y = 4 => 4 - 2y + 2y = 4 => 4 = 4. This is an identity, indicating infinitely many solutions.
- Row Echelon Form: Augmented matrix: [[1 1 | 2], [2 2 | 4]]. After row reduction: [[1 1 | 2], [0 0 | 0]]. The second row of zeros indicates infinitely many solutions.
Therefore, the system has infinitely many solutions.
By using these methods, you can effectively determine the number of solutions a linear system possesses.
