1.10 Systems of Simultaneous Linear Equations
1. Introduction
This section studies systems of equations, such as 2x2 systems ($a_1x + b_1y = c_1$) and 3x3 systems ($a_1x + b_1y + c_1z = d_1$).
For any $n \times n$ system, there are exactly three possibilities for the number of solutions:
- A unique solution (Consistent)
- No solution (Inconsistent)
- Infinitely many solutions (Consistent)
Example (1x1 System): For the simple equation $ax = b$:
- If $a \ne 0$: There is a unique solution $x = \dfrac{b}{a}$.
- If $a = 0$ and $b \ne 0$: There is no solution (e.g., $0x = 5$).
- If $a = 0$ and $b = 0$: There are infinitely many solutions ($0x = 0$ is true for any $x$).
2. 2x2 Systems
EXAMPLE 1
Solution set: $(1-2\lambda, \lambda), \text{ where } \lambda \in \mathbb{R}$.
3. 3x3 Systems
Gaussian Elimination can be used to solve these larger systems.
EXAMPLE 2
$x = 9 + 9\lambda, \quad y = -5 - 7\lambda, \quad z = \lambda$.
4. Gaussian Elimination
This method uses an augmented matrix to eliminate variables systematically without writing out the $x, y, z$ repeatedly.
Methodology:
- Write the augmented matrix of coefficients.
- Use row operations to create zeros below the main diagonal (upper triangular form).
- Interchange rows ($R_1 \leftrightarrow R_2$)
- Multiply a row by a non-zero scalar ($R_1 \to 5R_1$)
- Add a multiple of one row to another ($R_2 \to R_2 - 5R_1$)
- Analyze the final row to determine the nature of the solution, and use back-substitution.
REMARK: Ideally, during step 2, we attempt to create matrices where the leading coefficient of the selected row is $1$ to make row operations easier.
Solving System (a) Manually
Original System:
Step 1: Set up the augmented matrix. To get a leading $1$ at the top left, interchange $R_1$ and $R_2$.
Step 2: Eliminate $x$ from $R_2$ ($R_2 \to R_2 - 5R_1$) and $R_3$ ($R_3 \to R_3 - 3R_1$).
Step 3: Eliminate $y$ from $R_3$ ($R_3 \to R_3 + 8R_2$).
Step 4: Back Substitution
Solution: $(2, -1, 1)$.
Interpreting the Final Matrix
- Case 1 (Unique): Main diagonal is non-zero (like the example above).
-
Case 2 (No Solution): Last row takes the form $\left[ \begin{array}{ccc|c} 0 & 0 & 0 & d \end{array} \right]$ where $d \ne 0$.
This implies $0x + 0y + 0z = d$, which is mathematically impossible. -
Case 3 (Infinite Solutions): Last row takes the form $\left[ \begin{array}{ccc|c} 0 & 0 & 0 & 0 \end{array} \right]$.
This implies $0 = 0$ (which is always true). We must assign a free variable (e.g., $z = \lambda$) and solve for $x$ and $y$ in terms of $\lambda$.
5. System with Parameters
EXAMPLE 3
Suppose a sequence of Gaussian elimination row operations yields the following matrix:
If $a \ne 0$, there is a unique solution.
We calculate $z = \dfrac{b}{a}$, and then substitute back upwards to find exact numbers for $y$ and $x$.
If $a = 0$ and $b \ne 0$, the last row gives the impossible statement $0 = b$.
If $a = 0$ and $b = 0$, the last row reads $0 = 0$.
Set the free variable $z = \lambda$. General solution: $(-8 + 7\lambda, \, 6 - 5\lambda, \, \lambda)$.
⚠️ Geometrical Interpretation
- 2x2 System: Represents the intersection of two lines in a plane.
- 3x3 System: Represents the intersection of three planes in 3D space.