Section 5.4: Solving Special Systems of Linear Equations

Property

A pair of linear equations in two variables

considered together is called a system of linear equations. A solution is an ordered pair $(x, y)$ that satisfies each equation in the system. There are three types of linear systems:

1. Consistent and independent system. The graphs of the two lines intersect in exactly one point. The system has exactly one solution.
2. Inconsistent system. The graphs of the equations are parallel lines and hence do not intersect. An inconsistent system has no solutions.
3. Dependent system. The graphs of the two equations are the same line. A dependent system has infinitely many solutions.

Examples

• A consistent system like $y=x+2$ and $y=-x+4$ has one solution, $(1,3)$, where the two lines intersect.
• An inconsistent system like $y=3x+1$ and $y=3x+5$ has no solution. The lines have the same slope, so they are parallel and never cross.
• A dependent system like $x+y=5$ and $2x+2y=10$ has infinite solutions. The second equation is just the first one multiplied by 2, so they represent the same line.

Explanation

Think of each equation as a line on a graph. The system's solution is where these lines meet. They can cross at one point, run parallel and never touch, or be the exact same line, lying on top of each other.

Property

If two linear equations have the exact same slope (m), they will never intersect just once. You must check their y-intercepts (b) to determine the outcome:

• Different b (No Solution): The system has the same slope but different intercepts. They are parallel, so there is no solution.
• Same b (Infinite Solutions): The system has the same slope and the same y-intercept. They overlap everywhere, giving infinitely many solutions.

Examples

• No Solution: The system $y = 3x + 2$ and $y = 3x - 1$ has the same slope ($m=3$) but different $y$-intercepts. The lines are parallel, so there is no solution.
• Infinite Solutions: The system $x + y = 5$ and $2x + 2y = 10$ represents the same line, as the second equation is double the first. There are infinitely many solutions.

Explanation

When two lines have the same slope, they are traveling in the exact same direction at the exact same speed. If they start at different points on the y-axis, they will run parallel forever and never touch (zero solutions). But if they have the same slope AND start at the exact same y-intercept, they are actually a single line disguised as two different equations, meaning every point on the line is a shared solution (infinite solutions)!