Question

For what value of k, the following pair of linear equations has infinitely many solutions?

10x + 5y – (k – 5) = 0

20x + 10y – k = 0

Answer 1

Given: 10x + 5y – (k – 5) = 0, 20x + 10y – k = 0.

Step-wise calculation:

1. For infinitely many solutions the three ratios must be equal:

`a_1/a_2 = b_1/b_2 = c_1/c_2`

2. Identify coefficients:

a₁ = 10, b₁ = 5, c₁ = –(k – 5) = –k + 5 

a₂ = 20, b₂ = 10, c₂ = –k

3. Compute ratios:

`a_1/a_2 = 10/20 = 1/2`

`b_1/b_2 = 5/10 = 1/2` 

So require `c_1/c_2 = 1/2: (-k + 5)/(-k) = 1/2`

4. Solve: 2(–k + 5) = –k 

–2k + 10 = –k 

10 = k