System of linear equations and solutions
System of Linear Equations and Solutions
A system of linear equations consists of two or more linear equations involving the same set of variables. The solution to a system of linear equations is a set of values for the variables that makes all of the equations true simultaneously.
Representation of a System of Linear Equations
A system of linear equations can be written in matrix form as $AX = B$, where:
- $A$ is the coefficient matrix,
- $X$ is the column matrix of variables,
- $B$ is the column matrix of constants.
For example, consider the following system of linear equations:
[ \begin{align*} 2x + 3y &= 5 \ 4x - y &= 1 \end{align*} ]
In matrix form, this system is represented as:
[ \begin{bmatrix} 2 & 3 \ 4 & -1 \end{bmatrix} \begin{bmatrix} x \ y
\end{bmatrix}
\begin{bmatrix} 5 \ 1 \end{bmatrix} ]
Types of Solutions
A system of linear equations can have:
- A unique solution,
- No solution (inconsistent system),
- Infinitely many solutions.
The nature of the solutions depends on the rank of the coefficient matrix $A$ and the augmented matrix $[A|B]$.
| Type of Solution | Condition | Explanation |
|---|---|---|
| Unique Solution | Rank($A$) = Rank($[A | B]$) = Number of variables |
| No Solution | Rank($A$) < Rank($[A | B]$) |
| Infinitely Many Solutions | Rank($A$) = Rank($[A | B]$) < Number of variables |
Methods of Solving
There are several methods to solve a system of linear equations:
- Graphical Method
- Substitution Method
- Elimination Method
- Matrix Method (using inverse matrices)
- Cramer's Rule (if applicable)
- Gaussian Elimination
- Gauss-Jordan Elimination
Graphical Method
The graphical method involves plotting each equation on a graph and finding the point(s) of intersection.
Substitution Method
The substitution method involves solving one of the equations for one variable and substituting this expression into the other equations.
Elimination Method
The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variables.
Matrix Method
The matrix method involves finding the inverse of the coefficient matrix $A$ (if it exists) and then calculating $X = A^{-1}B$.
Cramer's Rule
Cramer's Rule can be used when the system has the same number of equations as variables, and the determinant of the coefficient matrix $A$ is non-zero. Each variable is found using determinants.
Gaussian Elimination
Gaussian Elimination involves transforming the augmented matrix $[A|B]$ into an upper triangular form and then performing back-substitution to find the solutions.
Gauss-Jordan Elimination
Gauss-Jordan Elimination involves transforming the augmented matrix $[A|B]$ into reduced row-echelon form, which directly provides the solutions.
Examples
Example 1: Unique Solution
Solve the following system of equations:
[ \begin{align*} x + 2y &= 5 \ 3x - y &= 4 \end{align*} ]
Using the elimination method:
[ \begin{align*} x + 2y &= 5 \quad \text{(1)} \ 3x - y &= 4 \quad \text{(2)} \end{align*} ]
Multiply equation (1) by 3 and equation (2) by 2:
[ \begin{align*} 3x + 6y &= 15 \quad \text{(3)} \ 6x - 2y &= 8 \quad \text{(4)} \end{align*} ]
Subtract equation (4) from equation (3):
[ \begin{align*} -3x + 8y &= 7 \ 8y &= 7 + 3x \ y &= \frac{7 + 3x}{8} \end{align*} ]
Substitute $y$ into equation (1):
[ \begin{align*} x + 2\left(\frac{7 + 3x}{8}\right) &= 5 \ x + \frac{7 + 3x}{4} &= 5 \ 4x + 7 + 3x &= 20 \ 7x &= 13 \ x &= \frac{13}{7} \end{align*} ]
Substitute $x$ back into equation (1) to find $y$:
[ \begin{align*} \frac{13}{7} + 2y &= 5 \ 2y &= 5 - \frac{13}{7} \ 2y &= \frac{35 - 13}{7} \ 2y &= \frac{22}{7} \ y &= \frac{11}{7} \end{align*} ]
The unique solution is $x = \frac{13}{7}$ and $y = \frac{11}{7}$.
Example 2: No Solution
Consider the following system of equations:
[ \begin{align*} x + y &= 2 \ x + y &= 5 \end{align*} ]
These two equations represent parallel lines that never intersect, hence there is no solution.
Example 3: Infinitely Many Solutions
Consider the following system of equations:
[ \begin{align*} x + 2y &= 4 \ 2x + 4y &= 8 \end{align*} ]
These two equations represent the same line, hence there are infinitely many solutions. Any point on the line is a solution to the system.
In summary, understanding the system of linear equations and their solutions is crucial for solving mathematical problems involving multiple variables. The methods of solving such systems vary depending on the nature of the equations and the number of solutions.