Number of Solutions to Systems of Equations

When dealing with a system of two linear equations, finding the solution means finding the point where the two lines intersect. Because lines are straight, there are only three possible ways they can interact on a graph. You can easily determine the number of solutions by comparing the slope ($m$) and the y-intercept ($b$) of the lines when written in slope-intercept form ($y = mx + b$).

The Three Possibilities

1. Exactly One Solution (Intersecting Lines) If the two lines have different slopes, they will eventually cross at exactly one point.

   • Rule: $m_1 \neq m_2$

2. No Solution (Parallel Lines) If the lines have the same slope but different y-intercepts, they are parallel. They will never cross, meaning the system has no solution.

   • Rule: $m_1 = m_2$ and $b_1 \neq b_2$

3. Infinitely Many Solutions (Same Line) If the lines have the same slope and the same y-intercept, they are literally the same line drawn on top of each other. Every point on the line is a shared solution.

   • Rule: $m_1 = m_2$ and $b_1 = b_2$

Checking the Number of Solutions

To figure out how many solutions a system has without graphing, simply rewrite both equations into slope-intercept form ($y = mx + b$) and compare their $m$ and $b$ values.

Example 1: Parallel Lines

Question: How many solutions does this system have? $y = 2x + 1$ $y = 2x + 3$

Solution: Both equations are already in $y = mx + b$ form.

• The slope of the first line is $2$, and the slope of the second line is $2$.
• The y-intercepts are $1$ and $3$.

Since the slopes are the same but the y-intercepts are different, these lines are parallel. Therefore, there is no solution.

Example 2: The Same Line

Question: How many solutions does this system have? $y = 3x - 2$ $6x - 2y = 4$

Solution: The first equation is in slope-intercept form, but the second one is not. Let's rewrite $6x - 2y = 4$:

Subtract $6x$ from both sides: $-2y = -6x + 4$

Divide every term by $-2$: $y = 3x - 2$

Now compare the two equations:

1. $y = 3x - 2$
2. $y = 3x - 2$

They have the exact same slope ($3$) and the exact same y-intercept ($-2$). Because they are the identical line, there are infinitely many solutions.