Systems of Linear Equations: Methods, Geometry & Applications

TAKEAWAYS:

• Three core methods (substitution, elimination, matrix) solve any system of linear equations, each with trade-offs in speed and complexity.
• Geometric interpretation: Solutions correspond to intersection points of lines/planes (unique, infinite, or no solution).
• Cramer’s rule uses determinants to find exact solutions but is limited to square systems with non-zero determinant.
• Applications span traffic routing (Pathao), budget allocation (Ncell), and loan amortization (Nepal Bank).
• Matrix inversion and Gaussian elimination are powerful for large systems but require careful arithmetic.
• Consistency checks (rank of augmented matrix) determine whether a system has solutions before attempting to solve it.

1. Definitions and Types of Systems

A system of linear equations is a collection of equations with the same variables, where each equation is linear (no exponents or products of variables). For example: $$\{\begin{array}{l}2x+3y=5\\ 4x-y=1\end{array}$$ Types of solutions:

• Unique solution: Lines intersect at one point (independent system).
• Infinite solutions: Lines coincide (dependent system).
• No solution: Parallel lines (inconsistent system).

Real-world analogy:

• Unique solution: A Daraz order with one delivery route (one intersection of constraints).
• No solution: A Kathmandu traffic route where two paths are blocked (parallel lines).
• Infinite solutions: A budget where any combination of items fits (coincident lines).

2. Methods to Solve Systems

A. Substitution Method

1. Solve one equation for one variable (e.g., $y=\frac{5-2x}{3}$).
2. Substitute into the other equation.
3. Solve for the remaining variable and back-substitute.

Worked Example 1: Solve: $$\{\begin{array}{l}x+2y=8\\ 3x-y=5\end{array}$$ Steps:

1. From equation (1): $x=8-2y$.
2. Substitute into (2): $3(8-2y)-y=5$.
3. Simplify: $24-6y-y=5$ → $-7y=-19$ → $y=\frac{19}{7}$.
4. Back-substitute: $x=8-2(\frac{19}{7})=\frac{16}{7}$.

Solution: $\left(\frac{16}{7},\frac{19}{7}\right)$.

When to use: Small systems (2–3 equations) where one variable is easily isolated.

B. Elimination Method

1. Align coefficients of one variable.
2. Add/subtract equations to eliminate that variable.
3. Solve the resulting equation and back-substitute.

Worked Example 2 (Real-world tie-in): A Nepal Bank offers two loan plans:

• Plan A: 10% interest + ₹500 fee → Total = $0.10x+500$.
• Plan B: 8% interest + ₹1000 fee → Total = $0.08x+1000$. Find the loan amount $x$ where both plans cost the same. System: $$\{\begin{array}{l}0.10x+500=0.08x+1000\\ \text{(Assume total cost equals for some }x\text{)}\end{array}$$ Steps:

1. Subtract equation (2) from (1): $0.02x-500=0$.
2. Solve: $x=25,000$.

Solution: At ₹25,000, both plans cost ₹2,500.

Advantages:

• Works for any system size.
• Avoids fractions (useful for integer coefficients).

C. Matrix Methods (Cramer’s Rule and Inversion)

For a system $A\mathbf{x}=\mathbf{b}$, where $A$ is the coefficient matrix:

• Cramer’s Rule: $x_i=\frac{\det (A_i)}{\det (A)}$, where $A_i$ replaces column $i$ of $A$ with $\mathbf{b}$.
• Matrix Inversion: $\mathbf{x}=A^{-1}\mathbf{b}$ (requires $\det (A)\ne 0$).

Worked Example 3: Solve using Cramer’s Rule: $$\{\begin{array}{l}2x+y=3\\ 4x-y=1\end{array}$$ Steps:

1. Compute $\det (A)$: $$A=\left[\begin{array}{cc}2& 1\\ 4& -1\end{array}\right],\det (A)=(2)(-1)-(1)(4)=-6.$$
2. Compute $\det (A_1)$ (replace column 1 with $\mathbf{b}$): $$A_1=\left[\begin{array}{cc}3& 1\\ 1& -1\end{array}\right],\det (A_1)=(3)(-1)-(1)(1)=-4.$$
3. Compute $\det (A_2)$ (replace column 2 with $\mathbf{b}$): $$A_2=\left[\begin{array}{cc}2& 3\\ 4& 1\end{array}\right],\det (A_2)=(2)(1)-(3)(4)=-10.$$
4. Solve: $$x=\frac{\det (A_1)}{\det (A)}=\frac{-4}{-6}=\frac{2}{3},y=\frac{\det (A_2)}{\det (A)}=\frac{-10}{-6}=\frac{5}{3}.$$

Solution: $\left(\frac{2}{3},\frac{5}{3}\right)$.

When to use:

• Cramer’s Rule: Small systems (≤3 equations) where determinants are easy to compute.
• Matrix Inversion: Programming (e.g., solving linear regression in Python).

D. Gaussian Elimination

Convert the augmented matrix $[A|\mathbf{b}]$ to row-echelon form using row operations:

1. Swap rows.
2. Multiply rows by non-zero scalars.
3. Add/subtract rows to create zeros below the pivot.

Worked Example 4: Solve: $$\{\begin{array}{l}x+2y-z=3\\ 2x-y+3z=7\\ 3x+y+z=4\end{array}$$ Augmented Matrix: $$\left[\begin{array}{ccccc}1& 2& -1& |& 3\\ 2& -1& 3& |& 7\\ 3& 1& 1& |& 4\end{array}\right]$$ Steps:

1. $R_2\leftarrow R_2-2R_1$, $R_3\leftarrow R_3-3R_1$: $$\left[\begin{array}{ccccc}1& 2& -1& |& 3\\ 0& -5& 5& |& 1\\ 0& -5& 4& |& -5\end{array}\right]$$
2. $R_3\leftarrow R_3-R_2$: $$\left[\begin{array}{ccccc}1& 2& -1& |& 3\\ 0& -5& 5& |& 1\\ 0& 0& -1& |& -6\end{array}\right]$$
3. Back-substitute:
   • From $R_3$: $z=6$.
   • From $R_2$: $-5y+5(6)=1$ → $y=\frac{29}{5}$.
   • From $R_1$: $x+2(\frac{29}{5})-6=3$ → $x=-\frac{13}{5}$.

Solution: $\left(-\frac{13}{5},\frac{29}{5},6\right)$.

flowchart TD
    A["Augmented Matrix"] --> B["R2 ← R2 - 2R1"]
    B --> C["R3 ← R3 - 3R1"]
    C --> D["R3 ← R3 - R2"]
    D --> E["Back-substitute for z, y, x"]

Advantages:

• Systematic for large systems (used in engineering simulations).
• Handles non-square systems (e.g., overdetermined systems in data fitting).

3. Geometric Interpretation

• 2D: Lines intersect at the solution point.
• 3D: Planes intersect at a line (infinite solutions) or a point (unique solution).
• No intersection: Parallel lines/planes (no solution).

Real-world example:

• Pathao’s route optimization: Solves a system where constraints (time, distance, traffic) are planes, and the solution is the fastest route (intersection point).

4. Consistency and Dependence

A system is:

• Consistent: Has at least one solution (lines/planes intersect).
• Inconsistent: No solution (parallel lines).
• Dependent: Infinite solutions (coincident lines).

Check using ranks:

• If $\text{rank}(A)=\text{rank}([A|\mathbf{b}])=n$: Unique solution.
• If $\text{rank}(A)=\text{rank}([A|\mathbf{b}])<n$: Infinite solutions.
• If $\text{rank}(A)\ne \text{rank}([A|\mathbf{b}])$: No solution.

Worked Example 5: Check consistency: $$\{\begin{array}{l}x+y=2\\ 2x+2y=4\\ x-y=0\end{array}$$ Augmented Matrix: $$\left[\begin{array}{cccc}1& 1& |& 2\\ 2& 2& |& 4\\ 1& -1& |& 0\end{array}\right]$$ Row Reduction:

1. $R_2\leftarrow R_2-2R_1$, $R_3$ remains: $$\left[\begin{array}{cccc}1& 1& |& 2\\ 0& 0& |& 0\\ 1& -1& |& 0\end{array}\right]$$
2. $R_3\leftarrow R_3-R_1$: $$\left[\begin{array}{cccc}1& 1& |& 2\\ 0& 0& |& 0\\ 0& -2& |& -2\end{array}\right]$$
3. $R_3\leftarrow R_3/-2$: $$\left[\begin{array}{cccc}1& 1& |& 2\\ 0& 0& |& 0\\ 0& 1& |& 1\end{array}\right]$$ Conclusion:

• $\text{rank}(A)=2$, $\text{rank}([A|\mathbf{b}])=2$, but the system reduces to $y=1$ and $x+1=2$ → $x=1$.
• Infinite solutions: All points $(1,1)$ satisfy the first two equations (dependent).

5. Applications in the Real World

A. Optimization Problems (e.g., Ncell’s Budget Allocation)

Ncell allocates budgets to marketing (M) and network expansion (N) with constraints: $$\{\begin{array}{l}M+N\le 100\text{ (total budget)}\\ 2M+N\le 150\text{ (marketing cap)}\\ M,N\ge 0\end{array}$$ Solution: Graph the constraints to find feasible region vertices (corner points), then evaluate objective function (e.g., maximize profit).

B. Traffic Flow (e.g., Kathmandu’s Ring Road)

Model traffic as a system where:

• Variables: Flow rates $x,y,z$ on different roads.
• Constraints: Conservation of flow (e.g., $x+y=z$ at an intersection). Example: $$\{\begin{array}{l}x+y=1000\text{ (total inflow)}\\ x-z=200\text{ (net outflow)}\\ y+z=800\text{ (outflow constraint)}\end{array}$$ Solution: Use elimination to find $x=600$, $y=400$, $z=400$.

C. Financial Modeling (e.g., NEPSE Stock Prices)

Predict stock prices using linear regression (a system of normal equations): $$A^{T}A\mathbf{w}=A^T\mathbf{y},$$ where $A$ is the design matrix (time points), $\mathbf{y}$ is stock prices, and $\mathbf{w}$ is the weight vector.

6. Comparison of Methods

7. Common Mistakes and Pitfalls

1. Forgetting to check consistency: Always verify $\text{rank}(A)=\text{rank}([A|\mathbf{b}])$.
2. Arithmetic errors: Double-check row operations in Gaussian elimination.
3. Misapplying Cramer’s Rule: Only use for square systems with $\det (A)\ne 0$.
4. Graphical inaccuracies: Ensure scales are consistent when plotting lines.

Example of a trap: Solve: $$\{\begin{array}{l}x+y=1\\ x+y=2\end{array}$$ Mistake: Assuming a solution exists (parallel lines). Correct Approach: Recognize inconsistency via $\text{rank}(A)=1\ne \text{rank}([A|\mathbf{b}])=2$.

## In the Real World

1. Pathao’s Route Optimization:

   • Idea: System of linear equations models traffic constraints (time, distance, capacity).
   • How: Solves for the fastest route by finding the intersection of feasible paths (lines in a 3D space of time, distance, cost).

2. Nepal Bank’s Loan Amortization:

   • Idea: Monthly payments $P$ and interest rates form a system where: $$P=\frac{r{P}_0}{1-(1+r{)}^{-n}},$$ with constraints on $P_0$ (loan amount) and $n$ (terms).
   • How: Banks use Gaussian elimination to solve for $P$ given $P_0$, $r$, and $n$.

3. Daraz’s Order Fulfillment Queue:

   • Idea: Orders are prioritized based on constraints (delivery time, warehouse stock).
   • How: A system of inequalities (e.g., $x+y\le \text{stock}$, $x\ge \text{min orders}$) is solved to maximize deliveries.

## Exam Tip

1. Always show steps: Partial credit is given for correct intermediate steps, even if the final answer is wrong.
2. Graphical methods for 2D systems: Plot lines to visualize solutions (especially for consistency checks).
3. Matrix methods for 3+ equations: Use Gaussian elimination or Cramer’s Rule (but verify $\det (A)\ne 0$ first).
4. Watch for hidden constraints: In word problems, translate phrases like "twice as much" into equations carefully.
5. Practice rank checks: For systems with no obvious solution, compute ranks to determine consistency.
6. Real-world applications: Expect questions tying systems to optimization (e.g., "Minimize cost under constraints").

High-scoring strategy:

• For substitution/elimination: Box each step (e.g., "From equation (1): $y=\dots$").
• For matrices: Label each row operation (e.g., "$R_2\leftarrow 3R_2+R_1$").
• For geometry: Sketch the system and label intersection points/solutions.

Visual Summary:

mindmap
  root((System of Linear Equations))
    Methods
      Substitution
      Elimination
      Matrix Methods
        Cramer's Rule
        Gaussian Elimination
    Geometric Interpretation
      2D: Lines
      3D: Planes
    Applications
      Optimization
      Traffic Flow
      Financial Modeling
    Consistency
      Unique Solution
      Infinite Solutions
      No Solution