For what value of K does the following system have an infinite number of solutions?

x + 2Ky - 8 = 0
2x - y - 16 = 0

Question

For what value of K does the following system have an infinite number of solutions?

x + 2Ky - 8 = 0
2x - y - 16 = 0

Accepted Answer

Given the system:
1) x + 2Ky - 8 = 0
2) 2x - y - 16 = 0

Rewrite equations in standard form:
1) x + 2Ky = 8
2) 2x - y = 16

For infinite solutions, the two equations must be dependent, i.e., one is a scalar multiple of the other.

Compare coefficients:
From equation 1: coefficients are (1, 2K, 8)
From equation 2: coefficients are (2, -1, 16)

Set ratios equal:
(1)/2 = (2K)/(-1) = 8/16

Calculate each ratio:
1/2 = 0.5
8/16 = 0.5

So, (2K)/(-1) must also be 0.5:
2K / (-1) = 0.5
=> 2K = -0.5
=> K = -0.5 / 2 = -0.25

But -0.25 = -1/4

Therefore, K = -1/4

Check options: -1/4 is option 3.

Re-examining the ratios shows a mistake: The ratios must be equal for all three parts.

Let's check carefully:

Ratio of x coefficients: 1 / 2 = 0.5
Ratio of y coefficients: 2K / (-1) = -2K
Ratio of constants: 8 / 16 = 0.5

For infinite solutions:
1/2 = 2K / (-1) = 8/16
=> 0.5 = -2K = 0.5

From -2K = 0.5
=> K = -0.25 = -1/4

Hence, the correct value of K is -1/4.

Therefore, the correct answer is option 3: -1/4.

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