Solving Systems of Linear Equations: Special Cases/Dependent 3.1

Systems of Linear Equations

Types of Solutions for Systems of Linear Equations

In College Algebra, a system of linear equations consists of two or more linear equations involving the same set of variables. The solution to the system is the set of all variable values that satisfy every equation in the system simultaneously. There are three main types of solutions:

• One Solution (Independent): The system has exactly one solution. The lines intersect at a single point.

• No Solution (Inconsistent): The system has no solution. The lines are parallel and never intersect.

• Infinitely Many Solutions (Dependent): The system has infinitely many solutions. The lines coincide (are the same line).

Solving Systems: Examples and Methods

Example 1: Infinite Solutions (Dependent System)

• Given: $2x - 4y = 3$ $6x - 12y = 9$

• Process: Multiply the first equation by 3: $3(2x - 4y) = 3 \times 3$ $6x - 12y = 9$ Subtract the second equation: $6x - 12y - (6x - 12y) = 9 - 9$ $0 = 0$ (True statement)

• Interpretation: The equations are equivalent, so there are infinitely many solutions. The solution set is all points on the line $2x - 4y = 3$.

• Solution Set: $\{(x, y) \mid 2x - 4y = 3\}$

• Expressing $y$ in terms of $x$: $2x - 4y = 3$ $-4y = -2x + 3$ $y = \frac{1}{2}x - \frac{3}{4}$

Example 2: No Solution (Parallel Lines)

• Given: $2x - 3y = 3$ $4x - 6y = -6$

• Process: Divide the second equation by 2: $2x - 3y = -3$ Compare with the first equation: $2x - 3y = 3$ The left sides are the same, but the right sides are different ($3 \neq -3$).

• Interpretation: The lines are parallel and never intersect. There is no solution.

• Expressing in slope-intercept form: $2x - 3y = 3 \implies y = \frac{2}{3}x - 1$ $2x - 3y = -3 \implies y = \frac{2}{3}x + 1$

• Conclusion: Both lines have the same slope ($\frac{2}{3}$) but different y-intercepts ($-1$ and $1$), confirming they are parallel.

Example 3: Infinite Solutions (Identity)

• Given: $y = -\frac{1}{5}x - \frac{4}{5}$ $y = -\frac{1}{5}x - \frac{4}{5}$

• Interpretation: Both equations are identical, so every point on the line is a solution. There are infinitely many solutions.

Summary Table: Types of Solutions for Systems of Linear Equations

Type	Algebraic Condition	Graphical Representation	Number of Solutions
Independent	Different slopes	Intersecting lines	One
Inconsistent	Same slope, different y-intercepts	Parallel lines	None
Dependent	Same slope, same y-intercept	Coinciding lines	Infinitely many

Key Points and Properties

• If both variables are eliminated and the resulting statement is true (e.g., $0 = 0$), the system has infinitely many solutions (dependent).

• If both variables are eliminated and the resulting statement is false (e.g., $0 = 5$), the system has no solution (inconsistent).

• If a unique solution is found (e.g., $x = a$, $y = b$), the system is independent.

Additional info:

• These concepts are foundational for understanding more advanced algebraic topics, such as matrices and determinants.

• Graphical interpretation helps visualize the nature of solutions for systems of equations.