You can tell how many solutions a system of linear equations has by comparing the slopes and y-intercepts of the lines they represent.
Understanding the relationship between two linear equations is key to determining their solution set. Each linear equation represents a straight line when graphed. The solution(s) to a system of two linear equations are the point(s) where the lines intersect.
Analyzing Linear Systems by Slope and Y-Intercept
For a system of two linear equations, typically written in slope-intercept form ($y = mx + b$), where '$m$' is the slope and '$b$' is the y-intercept, you can determine the number of solutions by comparing these values.
Based on the properties of the lines:
- If the slopes are different: The lines will intersect at exactly one point.
- If the slopes are the same: The lines are parallel. There are two possibilities here:
- If the y-intercepts are also the same: The lines are the exact same line (coincident).
- If the y-intercepts are different: The lines are distinct parallel lines.
This leads to the classification of solutions as described in the reference:
Case 1: One Solution
- Condition: The slopes of the two lines are different.
- Graphical Interpretation: The lines intersect at a single point.
- Number of Solutions: Exactly one solution.
Example:
A system like:
$y = 2x + 1$ (Slope = 2)
$y = -x + 3$ (Slope = -1)
Since the slopes (2 and -1) are different, this system has one solution.
Case 2: No Solution
- Condition: The slopes are the same, but the y-intercepts are different.
- Graphical Interpretation: The lines are parallel and never intersect.
- Number of Solutions: No solution.
As stated in the reference: "If the slopes are the same but the y-intercepts are different, the system has no solution."
Example:
A system like:
$y = 3x - 2$ (Slope = 3, Y-intercept = -2)
$y = 3x + 5$ (Slope = 3, Y-intercept = 5)
The slopes are the same (3), but the y-intercepts (-2 and 5) are different. These parallel lines will never meet.
Case 3: Infinitely Many Solutions
- Condition: The slopes are the same, and the y-intercepts are the same.
- Graphical Interpretation: The two equations represent the exact same line. Every point on the line is a solution.
- Number of Solutions: Infinitely many solutions.
As stated in the reference: "If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions."
Example:
A system like:
$y = 0.5x + 4$ (Slope = 0.5, Y-intercept = 4)
$2y = x + 8$ (Rewrite as $y = 0.5x + 4$; Slope = 0.5, Y-intercept = 4)
Both equations represent the identical line.
Summary Table
Here's a quick summary based on slope and y-intercept comparison for a system of two linear equations:
| Condition | Graphical Representation | Number of Solutions |
|---|---|---|
| Different Slopes | Intersecting Lines | One Solution |
| Same Slopes, Different Y-intercepts | Parallel Lines | No Solution |
| Same Slopes, Same Y-intercepts | Coincident Lines | Infinitely Many Solutions |
To apply this, ensure your equations are in slope-intercept form ($y = mx + b$) or determine the slopes and y-intercepts by rearranging the equations.
