Lesson 6-2: Determine Number of Solutions

Property

Before comparing two lines, you must write their equations in the slope-intercept form:

In this form, m represents the slope (rate of change), and b represents the y-coordinate of the y-intercept (0, b). By using algebraic manipulation to isolate y on one side, you can rewrite any linear equation into this format.

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 lines have different slopes, they will intersect at exactly one coordinate point (x, y). To find this exact point algebraically, set their two expressions equal to each other:

Solve for x, then substitute that value back into either original equation to find y.

Examples

• Find the intersection of $y = 2x + 3$ and $y = -x + 6$.
  • Step 1 (Set Equal): $2x + 3 = -x + 6$.
  • Step 2 (Solve for x): Add x to both sides ($3x + 3 = 6$), then subtract 3 ($3x = 3$), which gives $x = 1$.
  • Step 3 (Find y): Substitute $x = 1$ into the first equation: $y = 2(1) + 3 = 5$. The exact intersection point is $(1, 5)$.

Explanation

If two functions have different rates of change (slopes), they are guaranteed to crash into each other exactly once. Because they share the exact same y-value at the moment they crash, you can set their equations equal to each other. This creates a simple one-variable puzzle to find the x-coordinate of the crash site! Once you find the x-value, substitute it back into either of the original equations to calculate the corresponding y-value.