Which of the following is the sum of the values of a and b if the equations 2x + y = a, 8x + by = 12 have infinite solutions?

Question

Accepted Answer

For two linear equations to have infinite solutions, they must be proportional to each other.

Given equations:
1) 2x + y = a
2) 8x + by = 12

Compare coefficients:
The ratio of coefficients of x in both equations = 2/8 = 1/4
The ratio of coefficients of y in both equations = 1/b
The ratio of constants = a/12

For infinite solutions, all these ratios must be equal:
1/4 = 1/b = a/12

From 1/4 = 1/b, we get b = 4
From 1/4 = a/12, cross-multiplying gives 4a = 12 => a = 3

Sum of a and b = 3 + 4 = 7

Let's re-check the ratios carefully.

Actually, ratio of x coefficients: 2/8 = 1/4
Ratio of y coefficients: 1/b
Ratio of constants: a/12

Set 1/4 = 1/b => b = 4
Set 1/4 = a/12 => cross multiply: 12 * 1 = 4 * a => 12 = 4a => a = 3
Sum = a + b = 3 + 4 = 7

Option 3 is 7, so option 3 is correct.

Therefore, the sum of a and b is 7.

Answer

Answer

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