Identify systems with one solution

A system of two linear equations in two variables has one solution when the equations represent distinct lines that intersect at exactly one point, equivalently when their slopes are different (including cases involving a vertical line). The solution is the single ordered pair whose coordinates satisfy both equations, whether identified from a graph or determined algebraically; parallel lines have no solution, while identical lines have infinitely many, cases that distinguish unique solutions from other outcomes.

Detailed Explanation: Identify systems with one solution

A system of two linear equations has one solution when the two lines have different slopes. Different slopes mean the lines intersect at exactly one point.

Consider the system:

\begin{cases} y=2x+1\\ y=-x+4 \end{cases}

Step 1: Compare the slopes

Both equations are in the form $y=mx+b$ , where $m$ is the slope.

The first line has slope $2$ .
The second line has slope $-1$ .

Because $2\neq -1$ , the lines are not parallel. They must intersect at exactly one point.

Step 2: Find the intersection point

At the intersection, both equations have the same $y$ -value, so set the right sides equal:

2x+1=-x+4

Add $x$ to both sides:

3x+1=4

Subtract $1$ :

3x=3

Divide by $3$ :

x=1

Substitute $x=1$ into either equation:

y=2(1)+1=3

So the system has one solution:

\boxed{(1,3)}

The lines intersect at the point $(1,3)$ .

Remember:

Different slopes $\rightarrow$ one solution
Same slope but different lines $\rightarrow$ no solution
Same line $\rightarrow$ infinitely many solutions

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Types of Solutions - Equation Pair to Zero, One, or Infinite

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