# Ex.4.2 Q1 Linear Equations in Two Variables Solution - NCERT Maths Class 9

## 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

## Text Solution

**What is known?**

Linear equation \(\begin{align} y = 3x + 5\end{align} \)

**What is Unknown?**

Number of solutions of the given equation.

**Reasoning: **

We can check number of solution by putting different values of \(x\) and get different values of \(y\).

**Steps:**

**We know that**

- \(y = 3x + 5\) is a linear equation in two variables in the form of \(ax + by + c = 0\)
- For \(x = 0, y = 0 + 5 = 5\) Therefore, \((0, 5)\) is one solution.
- For \(x = 1, y = 3×1 + 5 = 8\) Therefore, \((1, 8)\) is another solution.
- For \(y = 0, 3x + 5 = 0, x = - 5/3\) Therefore, \((- 5/3 , 0)\) is another solution .

Clearly, for different values of \(x\), we get various values for \(y\).

Thus, any value substituted for \(x\) in the given equation will constitute another solution for the given equation.

So, there is no end to the number of different solutions obtained on substituting real values for \(x\) in the given linear equation. Therefore, a linear equation in two variables has infinitely many solutions.

Hence (iii) is the correct answer