Systems of Linear Equations and Their Solutions

5. Systems of Linear Equations and Their Solutions

5.1 What is a System of Linear Equations?

A system of linear equations is a collection of one or more linear equations involving the same set of variables. The goal is to find values for these variables that satisfy all the equations simultaneously.

For example, the system of equations:
[
$$\begin{array}{rlr}x+y& =22x-y& =1\end{array}$$
]
is a system of two linear equations in two variables, $x$ and $y$.

5.2 Types of Solutions

There are three possible outcomes for a system of linear equations:
1. Unique solution: A single point where all equations intersect.
2. No solution: The system is inconsistent, meaning the equations contradict each other.
3. Infinitely many solutions: The system is dependent, meaning the equations describe the same line or plane.

5.3 Solving Systems of Linear Equations

There are several methods to solve systems of linear equations:

5.3.1 Substitution Method

1. Solve one equation for one variable.
2. Substitute this expression into the other equation(s).
3. Solve for the remaining variable(s).

Example:
[
$$\begin{array}{rlr}x+y& =2\text{(Equation 1)}2x-y& =1\text{(Equation 2)}\end{array}$$
]

1. Solve Equation 1 for $y$:
   [
   y = 2 - x
   ]
2. Substitute into Equation 2:
   [
   2x - (2 - x) = 1 \
   2x - 2 + x = 1 \
   3x = 3 \
   x = 1
   ]
3. Substitute $x=1$ back into Equation 1:
   [
   1 + y = 2 \quad \Rightarrow \quad y = 1
   ]

Solution: $x=1,y=1$.

---

5.3.2 Elimination (or Addition) Method

1. Multiply one or both equations by constants to align coefficients of one variable.
2. Add or subtract the equations to eliminate one variable.
3. Solve for the remaining variable.

Example:
[
$$\begin{array}{rlr}x+y& =2\text{(Equation 1)}2x-y& =1\text{(Equation 2)}\end{array}$$
]

1. Add Equation 1 and Equation 2 to eliminate $y$:
   [
   (x + y) + (2x - y) = 2 + 1 \
   3x = 3 \quad \Rightarrow \quad x = 1
   ]
2. Substitute $x=1$ back into Equation 1:
   [
   1 + y = 2 \quad \Rightarrow \quad y = 1
   ]

Solution: $x=1,y=1$.

---

5.3.3 Matrix Method (Gaussian Elimination)

This method involves converting the system of equations into an augmented matrix and performing row operations to solve for the variables.

Consider the system:
[
$$\begin{array}{rlr}x+y& =2\text{(Equation 1)}2x-y& =1\text{(Equation 2)}\end{array}$$
]
This system can be written as the augmented matrix:
[
$$\left[\begin{array}{ccccccc}1& 1& |& 22& -1& |& 1\end{array}\right]$$
]
The goal is to use row operations to simplify this matrix into row echelon form, and then use back substitution to solve.

Step 1: Subtract 2 times Row 1 from Row 2:
[
$$\left[\begin{array}{ccccccc}1& 1& |& 20& -3& |& -3\end{array}\right]$$
]

Step 2: Multiply Row 2 by $-\frac{1}{3}$:
[
$$\left[\begin{array}{ccccccc}1& 1& |& 20& 1& |& 1\end{array}\right]$$
]

Step 3: Subtract Row 2 from Row 1:
[
$$\left[\begin{array}{ccccccc}1& 0& |& 10& 1& |& 1\end{array}\right]$$
]

This corresponds to the system:
[
x = 1, \quad y = 1
]

Solution: $x=1,y=1$.

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5.4 Special Cases

1. Consistent and Independent System: The system has exactly one solution (unique solution).
2. Consistent and Dependent System: The system has infinitely many solutions, typically represented by a free variable.
3. Inconsistent System: The system has no solution because the equations contradict each other.

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Practice Problems

1. Solve the following system using substitution or elimination:
   [
   $$\begin{array}{rlr}3x+2y& =74x-y& =5\end{array}$$
   ]

2. Solve the system using the elimination method:
   [
   $$\begin{array}{rlr}x-2y& =43x+4y& =12\end{array}$$
   ]

3. Solve the following system using Gaussian elimination:
   [
   $$\begin{array}{rlrl}x+y+z& =62x-y+3z& =143x+2y+3z& =10\end{array}$$
   ]

4. Determine whether the following system has a solution:
   [
   $$\begin{array}{rlr}x+y& =32x+2y& =6\end{array}$$
   ]

Would you like solutions to these problems, or should we move to Eigenvalues and Eigenvectors next?

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