Question

Which of the following options is true?
\[y = 3x + 7\] has
A. A unique solution
B. Two solutions
C. Infinitely many solutions
D. No solution

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. 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,
\[y = 3x + 7 - - - (1)\]
We know that an equation tells us that two 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.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 the equation \[(1)\], we get
\[ \Rightarrow y = \left( {3 \times 1} \right) + 7\]
\[ \Rightarrow y = 3 + 7\]
On adding, we get
\[ \Rightarrow y = 10\]
Now, taking \[x = 2\] in the equation \[(1)\], we get
\[ \Rightarrow y = \left( {3 \times 2} \right) + 7\]
\[ \Rightarrow y = 6 + 7\]
On adding, we get
\[ \Rightarrow y = 13\]

Now, taking \[x = 3\] in the equation \[(1)\], we get
\[ \Rightarrow y = \left( {3 \times 3} \right) + 7\]
\[ \Rightarrow y = 9 + 7\]
On adding, we get
\[ \therefore y = 16\]
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. Therefore, \[y = 3x + 7\] have infinite possible solutions.

Hence, option (C) is correct.

Note:If we are required 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 ordered pairs. There are infinite positive integers.