Types of Solutions for Systems of Linear Equations (Unique, Infinite, None)
The Solution of System of Linear Equations is a core topic in algebra, essential for students preparing for school exams (Class 10, 12), competitive tests (like JEE), and those interested in how mathematics solves real-life problems. Understanding how to solve systems helps in topics ranging from physics to economics and technology.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations involving the same set of variables. The solution to a system is the set of variable values that makes every equation in the system true at the same time.
- The most common systems are in two variables (2x2) or three variables (3x3).
- Systems are used to solve real-world problems where several conditions must be satisfied together, such as budgeting, mixture problems, or finding points of intersection in geometry.
Types of Solutions in Systems of Equations
A system of linear equations can have:
- Unique Solution: The equations intersect at exactly one point. This is called a consistent and independent system.
- Infinite Solutions: The equations represent the same line (coincident), so every point on the line is a solution (consistent and dependent system).
- No Solution: The equations represent parallel lines that never meet (inconsistent system).
Geometrically, two lines on a graph will either intersect at a point (unique), lie on top of each other (infinite), or never meet (no solution).
Methods to Solve a System of Linear Equations
Several methods are used to solve systems of linear equations:
- Substitution Method: Solve one equation for one variable and substitute it in the other.
- Elimination Method: Add or subtract equations to eliminate one variable.
- Graphical Method: Graph each equation and find the intersection point(s).
- Matrix Methods: Use matrix algebra, including:
- Inverse Matrix Method
- Cramer’s Rule (uses determinants)
- Gauss Elimination (row operations)
The choice of method depends on the number of variables, the context of the problem, and your confidence with algebra or matrices.
Important Formulas for System of Linear Equations
Here are some crucial formulas and matrix representations:
- Matrix Form: \(AX = B\) where
- \(A\) is the coefficient matrix
- \(X\) is the column of variables
- \(B\) is the constant column
- Inverse Matrix Method: If \(A\) is invertible, then \(X = A^{-1}B\)
- Cramer’s Rule (2x2 Example):
| For system: \( a_1x + b_1y = c_1 \) \( a_2x + b_2y = c_2 \) |
\( x = \dfrac{\begin{vmatrix}c_1 & b_1\\c_2 & b_2 \end{vmatrix}}{\begin{vmatrix}a_1 & b_1\\a_2 & b_2 \end{vmatrix}} \) \( y = \dfrac{\begin{vmatrix}a_1 & c_1\\a_2 & c_2 \end{vmatrix}}{\begin{vmatrix}a_1 & b_1\\a_2 & b_2 \end{vmatrix}} \) |
Where \( \begin{vmatrix} \cdot \end{vmatrix} \) denotes the determinant.
Worked Example: Solving a 2x2 System by Elimination
Solve the system:
\( 2x + 3y = 13 \)
\( x - 2y = -4 \)
- Multiply the second equation by 2:
\( 2x - 4y = -8 \) - Subtract this from the first equation:
\( (2x + 3y) - (2x - 4y) = 13 - (-8) \)
\( 7y = 21 \)
\( y = 3 \) - Substitute \( y = 3 \) in \( x - 2y = -4 \):
\( x - 2(3) = -4 \implies x - 6 = -4 \implies x = 2 \) - Solution: \( x = 2, y = 3 \)
Practice Problems
- Solve by substitution: \( x + y = 7; \; 2x - y = 4 \)
- Use elimination: \( 3x + 2y = 11; \; 2x - y = 1 \)
- Express in matrix form and solve: \( x + 2y = 9; \; 4x - y = 5 \)
- Use Cramer’s Rule: \( 2x - 3y = 1; \; 5x + 2y = 12 \)
- If a system has no solution, what does it mean graphically?
Common Mistakes to Avoid
- Incorrectly aligning variables when using elimination or substitution.
- Arithmetic errors in adding/subtracting or in matrix calculations.
- Forgetting to check if the determinant is zero before applying Cramer’s Rule or inverse matrix method.
- Confusing unique, infinite, and no solution cases.
- Missing the geometric meaning: not graphing or visualizing for quick checks.
Real-World Applications
Systems of linear equations are used in budgeting (comparing two buying options), physics (motion with multiple forces), chemistry (mixing solutions), and business (solving for price and profit). For instance, in a mixture problem, you might have two unknowns—quantity and concentration—which you find using two equations.
Page Summary
In this topic, you learned about the solution of system of linear equations using various methods like substitution, elimination, and matrix techniques. Recognizing the solution type and avoiding common errors are crucial for quick and accurate problem-solving in both exams and real-life scenarios. At Vedantu, we break down each concept into simple steps, helping you master linear systems and excel in mathematics.
You may also like to explore related topics: Linear Equations in One Variable, Matrices, Cramer's Rule, Gauss Elimination Method.
FAQs on How to Solve a System of Linear Equations: Step-by-Step Guide
1. How do you find the solution to a system of linear equations?
Solving a system of linear equations involves finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, graphical methods, and matrix methods (like inverse matrix method, Cramer's Rule, and Gauss elimination). The best method depends on the specific system.
2. What is Cramer’s Rule, and when can I use it?
Cramer's Rule is a method for solving systems of linear equations using determinants. It's particularly useful for systems with a unique solution and is often used for 2x2 or 3x3 systems. It's not efficient for larger systems.
3. How do I solve a system using matrices?
Matrix methods offer a systematic way to solve systems of linear equations. First, represent the system in matrix form (AX = B). Then, depending on the system type, employ techniques like finding the inverse matrix (X = A-1B) or using row operations (like Gaussian elimination) to solve for the variables.
4. What is the difference between unique, infinite, and no solutions?
A system of linear equations can have one of three solution types: A unique solution means there's only one set of values satisfying all equations. Infinite solutions occur when the equations are linearly dependent, resulting in many solutions. No solution means the equations are inconsistent; they cannot be satisfied simultaneously. Graphically, these represent intersecting lines (unique), overlapping lines (infinite), and parallel lines (no solution).
5. Can a non-square matrix system be solved by the inverse matrix method?
No, the inverse matrix method requires a square matrix (same number of equations as variables). For non-square systems, other methods like Gaussian elimination are needed.
6. What are the different methods for solving systems of linear equations?
Several methods exist for solving systems of linear equations. Common techniques include: Substitution method, Elimination method, Graphical method, and Matrix methods (including the inverse matrix method and Cramer's rule).
7. How do you use matrices to solve systems of linear equations?
To solve using matrices, first, write the system in matrix form (AX=B). If matrix A is square and invertible, find the inverse of matrix A (A-1) and calculate X = A-1B, which will provide the solution.
8. What does “no solution” mean in a system of linear equations?
A “no solution” outcome means the equations in the system are inconsistent; there's no set of values that can satisfy all equations simultaneously. Graphically, this is represented by parallel lines.
9. What is the solution set of the system?
The solution set represents all possible values of the variables that satisfy the system of equations. It can be a unique solution, infinite solutions, or an empty set (no solution).
10. How do you solve a system of linear equations using the Gauss elimination method?
The Gauss elimination method uses row operations on the augmented matrix to transform it into row-echelon form. This allows you to systematically solve for the variables, starting from the last row and working backward.
11. What are some real-life applications of systems of linear equations?
Systems of linear equations have wide applications. Examples include analyzing network flows, solving circuit problems in electrical engineering, analyzing economic models, and determining the optimal mixture of ingredients in chemistry.
