Lesson 4: Solving Special Systems of Linear Equations

Property

A system of equations can be classified by the number of solutions, which is determined entirely by how the two lines relate to each other visually and algebraically:

• One Solution (Intersecting): The lines have different slopes. The system is consistent and independent.
• No Solution (Parallel): The lines have the exact same slope but different y-intercepts. The system is inconsistent.
• Infinite Solutions (Coincident): The lines have the same slope and the same y-intercept. The system is consistent and dependent.

Examples

• One Solution: The lines $y = x + 1$ and $y = -x + 3$ intersect at the point $(1, 2)$, giving exactly one solution.
• No Solution: The lines $y = 2x + 1$ and $y = 2x + 4$ are parallel (same slope, different y-intercepts) and never intersect.
• Infinite Solutions: The equations $y = 3x - 2$ and $6x - 2y = 4$ represent the same line when graphed, so every point on the line is a solution.

Explanation

When you graph two lines, they can only relate in three ways: they cross once, they never cross because they are parallel, or they are actually the exact same line. By simply looking at the 'm' and 'b' in their equations, you can instantly predict how many solutions the system has without even needing to draw the graph!

Property

When using substitution, the variables will sometimes completely cancel each other out. The resulting numerical statement determines the number of solutions and the geometric classification of the system:

• Identity (True Statement): If solving results in a true statement like $0 = 0$ or $5 = 5$, the system has Infinitely Many Solutions. The equations are dependent (they represent the exact same coincident line).
• Contradiction (False Statement): If solving results in a false statement like $0 = -10$ or $2 = -1$, the system has No Solution. The equations are inconsistent (they represent parallel lines).

Examples

• Infinite Solutions (Identity): Solve $y = 2x - 3$ and $4x - 2y = 6$.

Substitute $y$: $4x - 2(2x - 3) = 6$.
Distribute: $4x - 4x + 6 = 6$.
Simplify: $6 = 6$. This is a true statement, so there are infinitely many solutions.

• No Solution (Contradiction): Solve $y = 5x + 2$ and $y = 5x - 1$.

Substitute $y$: $5x + 2 = 5x - 1$.
Subtract $5x$ from both sides: $2 = -1$. This is a false statement, so there is no solution.

Explanation

If your variables vanish during substitution, the algebra is trying to tell you something about the geometry of the lines! A true statement like $6 = 6$ means the two equations are actually identical twins disguised as different math problems; they overlap perfectly. A false statement like $2 = -1$ means you have reached a mathematical dead-end because the lines are parallel and will literally never cross.

Property

If solving a system of equations by substitution results in a true statement without variables, such as $0=0$, the equations are dependent. The graphs of the two equations are the same line, and the system has infinitely many solutions.

Examples

• Solve the system $\begin{cases} y = 2x - 3 \\ 4x - 2y = 6 \end{cases}$. Substitute $y$ in the second equation: $4x - 2(2x-3) = 6$. This simplifies to $4x - 4x + 6 = 6$, which gives $6=6$. This is a true statement, so there are infinitely many solutions.

• Solve the system $\begin{cases} x = 3y + 1 \\ 2x - 6y = 2 \end{cases}$. Substitute $x$ in the second equation: $2(3y+1) - 6y = 2$. This simplifies to $6y + 2 - 6y = 2$, which results in $2=2$. This is a true statement, indicating infinitely many solutions.