Question

Find the value of k for which the following system of equations have infinitely many solution:

2x + 3y = 7 

(k + 1)x + (2k – 1)y = 4k + 1

Answer 1

Given: 2x + 3y = 7, (k + 1)x + (2k – 1)y = 4k + 1

Step-wise calculation:

1. Write both in standard form:

2x + 3y – 7 = 0 and (k + 1)x + (2k – 1)y – (4k + 1) = 0

2. For infinitely many solutions the ratios must be equal:

`a_1/a_2 = b_1/b_2 = c_1/c_2` 

So `2/(k + 1) = 3/(2k - 1) = 7/(4k + 1)`.

3. Solve `2/(k + 1) = 3/(2k - 1)`:

2(2k – 1) = 3(k + 1) 

⇒ 4k – 2 = 3k + 3

⇒ k = 5

4. Check the other ratio with k = 5:

`3/(2 xx 5 - 1) = 3/9 = 1/3` and `7/(4 xx 5 + 1) = 7/21 = 1/3`, so they match.