Question

Which one of the following options is true, and why? \[y = 3x + 5\] has
(i)a unique solution,
(ii)only two solutions,
(iii)infinitely many solutions

Answer 1

Hint: Here, we will use the meaning of solution, which tells us that the solution of the equation with two variables is a value of \[x\] and \[y\], when substituted into the equation, satisfies the equation. So we will take the different values of \[x\] to find the value of \[y\] in the given equation to find the number of solutions.

Complete step-by-step answer:
We are given that the equation is
\[y = 3x + 5{\text{ ......eq(1)}}\]
We know that an equation tells us that two of sides are equal with some variables and constants.
We also know that a solution of the equation with two variables is a value of \[x\] and \[y\], when substituted into the equation, satisfies the equation.
We will take the different values of \[x\] to find the value of \[y\] in the equation (1).
Let us first take \[x = 1\] in equation (1), we get
\[
   \Rightarrow y = 3\left( 1 \right) + 5 \\
   \Rightarrow y = 3 + 5 \\
   \Rightarrow y = 8 \\
 \]
Now taking \[x = 2\] in the equation (1), we get
\[
   \Rightarrow y = 3\left( 2 \right) + 5 \\
   \Rightarrow y = 6 + 5 \\
   \Rightarrow y = 11 \\
 \]
Taking \[x = 3\] in the equation (1), we get
\[
   \Rightarrow y = 3\left( 3 \right) + 5 \\
   \Rightarrow y = 9 + 5 \\
   \Rightarrow y = 14 \\
 \]
We can see that \[x\] can have infinite values and for infinite values of \[x\], there can be infinitely many values of \[y\] as well.
So, \[y = 3x + 5\] have infinite possible solutions.
Hence, option (iii) is correct.


Note: Students need to know that if you are asked to find the solution of just one equation with two variables that would mean that we have to find some value of the variable like \[y\] in the given equation by taking different values of \[x\]. The solution of a linear equation is always written in an ordered pair. One should know that the positive integers are infinite.