4.2: Number of Solutions | Algebra 1 Honors

How it works

Two lines may intersect once, be distinct and parallel, or coincide. These cases correspond to one solution, no solutions, or infinitely many solutions. In algebra, a false constant statement signals inconsistency, while an identity can signal equivalent equations.

0=5\Rightarrow\varnothing,\qquad 0=0\Rightarrow\text{check for equivalent equations}

Worked examples

Example 1

Classify $\displaystyle y=2x+1$ and $\displaystyle y=2x-3$ .

They have the same slope but different intercepts, so they are distinct parallel lines with no solution.

Example 2

Classify $\displaystyle x+y=4$ and $\displaystyle 2x+2y=8$ .

The second equation is twice the first. Every point on $\displaystyle x+y=4$ solves both equations, giving infinitely many solutions.

Problems

Problem 4.2.1

How many solutions do $\displaystyle y=-x+2$ and $\displaystyle y=3x-2$ have?

Show worked solution

The slopes differ, so there is one solution. Equating gives $\displaystyle 4x=4$ , hence the point $\displaystyle (1,1)$ .

Problem 4.2.2

Classify $\displaystyle 3x-y=2$ and $\displaystyle 6x-2y=9$ .

Show worked solution

Doubling the first gives $\displaystyle 6x-2y=4$ , incompatible with $\displaystyle 9$ . There is no solution.

Problem 4.2.3

Classify $\displaystyle x-2y=5$ and $\displaystyle 3x-6y=15$ .

Show worked solution

The equations are equivalent, so there are infinitely many solutions on the same line.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.2.1

Problem 4.2.2

Problem 4.2.3

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