For the pair of equations λ x + 3y = – 7, 2x – 6y = 14 to have infinitely manysolutions, find the value of λ

1. Break Down the Problem

To determine the value of $\lambda$ such that the system of equations has infinitely many solutions, we need to establish a relationship between the two equations. This generally occurs when one equation is a scalar multiple of the other.

2. Relevant Concepts

The equations are:

1. $\lambda x + 3y = -7$ (Equation 1)
2. $2x - 6y = 14$ (Equation 2)

For the equations to have infinitely many solutions, they must be equivalent (i.e., multiples of each other).

3. Analysis and Detail

First, we can express Equation 2 in a more usable form. Dividing the entire equation by 2 gives: $x - 3y = 7 \quad (Equation 2')$

Now, let's express Equation 2' in a comparable form with Equation 1. To check for equivalency, we rearrange Equation 1: $\lambda x + 3y + 7 = 0$

To be equivalent, the coefficients must maintain a constant ratio: $\frac{\lambda}{1} = \frac{3}{-3} = \frac{-7}{7}$ This leads to the following relationships: $\frac{\lambda}{1} = -1 \quad \text{and} \quad \frac{-7}{7} = -1$ From the first ratio, we find: $\lambda = -1$

4. Verify and Summarize

Let's verify if substituting $\lambda = -1$ makes the two equations equivalent:

1. If $\lambda = -1$, then Equation 1 becomes: $-x + 3y = -7 \quad \Rightarrow \quad x - 3y = 7$

This matches Equation 2'. Thus, both equations represent the same line, confirming they will have infinitely many solutions.

Final Answer

The value of $\lambda$ for which the equations have infinitely many solutions is: $\lambda = -1$