Question

The simultaneous equations $x + 2y = 70$ and $2x + \lambda y = 35$ have no solution if $\lambda$ will be:

• A. $\frac{1}{4}$
• B. $\frac{1}{2}$
• C. $2$
• D. $4$

Hint:

For no solution, the ratios of the coefficients of $x$ and $y$ must be equal, but the ratio of the constant terms should not match.

Answer 1

The Correct Option is D

Step 1: Recall the condition for no solution.
For a pair of linear equations to have no solution, the ratio of the coefficients of $x$ and $y$ must be equal, but the constant terms must not be in the same ratio. The condition for no solution is: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
Step 2: Apply the condition to the given equations.
The given equations are: \[ x + 2y = 70 \quad \text{and} \quad 2x + \lambda y = 35 \] Here, the coefficients are: \[ a_1 = 1, \, b_1 = 2, \, c_1 = 70 \] \[ a_2 = 2, \, b_2 = \lambda, \, c_2 = 35 \]
Step 3: Set up the equation for the ratio of the coefficients.
For no solution: \[ \frac{1}{2} = \frac{2}{\lambda} \]
Step 4: Solve for $\lambda$.
\[ \frac{1}{2} = \frac{2}{\lambda} \quad \Rightarrow \quad \lambda = 4 \]
Step 5: Conclusion.
Hence, the value of $\lambda$ is $4$.