Question

The equation $y=2x+3$has
A.Unique solutions
B.No solutions
C.Only two solutions
D.Infinitely many solutions

Answer 1

Hint: The given equation would have a unique solution if it has only one set of (x,y) that would satisfy the given equation. If there exists no answer to a particular equation then it would have no solution and if there are an infinite set of values that satisfy the equation then it would be said to have infinitely many solutions.

Complete step-by-step answer:
To find whether the above given equation has unique, no, only two or infinitely many solutions we have to check how many pairs of (x,y) would satisfy the above equation.
So if we take $x=0$in the given equation $y=2x+3$
$\begin{align}
  & y=2(0)+3 \\
 & y=0+3 \\
 & y=3 \\
\end{align}$
So (0,3) would satisfy the given equation.
Since (0,3) satisfies the above given equation, so option (B) Unique solution cannot be the answer.
Similarly, when we take $x=1$
$\begin{align}
  & y=2(1)+3 \\
 & y=2+3 \\
 & y=5 \\
\end{align}$
So (1,5) satisfy the given equation.
As there are two pairs of (x,y) i.e (0,3) and (1,5) that satisfies the above equation, so option (A) unique solution also cannot be the answer.
Similarly when we take $x=2$
$\begin{align}
  & y=2(2)+3 \\
 & y=4+3 \\
 & y=7 \\
\end{align}$
So (2,7) also satisfies the given equation.
As (0,3), (1,5), (2,7) satisfy the above equation, so option (C) only two solutions also cannot be the answer.
As for infinite values of x we would have infinite values of y. Hence an infinite pair of (x,y) would satisfy the above equation.
So the correct answer is option (D) infinitely many solutions.
So, the correct answer is “Option D”.

Note: If we plot the above equation on the graph we would get a straight line on the graph which depicts that the given equation has infinitely many solutions. Thus a given equation would have infinitely many solutions if there are an infinite pair of (x,y) that satisfy a particular equation.