Question

Can a system of two linear equations have exactly two solutions? Why or why not?

Answer 1

A system of linear equations in two variables may have zero, one, two, or infinitely many solutions:

Any linear equation has a graph with a straight line.

Consider a system of two linear equations, then we get three possibilities.

That are,

(i) A system of two linear equations may have two parallel lines.

For example:

$$\begin{gathered} y=2x+1 \\ y=2x-1 \end{gathered}$$

Since, the two linear equation gives the same slope but different intercepts.

(ii) A system of two linear equations may intersect each other.

$$\begin{gathered} y=5x-2 \\ y=2x+1 \end{gathered}$$

That is the two different linear equations are crossing lines also we know that two straight lines cannot meet at two points.

(iii) A system of two linear equations may have the same line as their graph.

$$\begin{gathered} x+y=4 \\ x+y=4 \end{gathered}$$

Here, the lines coincide each other, then the solution is called trivial solution.

Hence, no system of linear equations in two variables have exactly two solutions.