Question

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x + 3y = 7

2ax + ay = 28 – by

Answer 1

Given: 2x + 3y = 7, 2ax + ay = 28 – by

Step-wise calculation:

1. Put both equations in standard form (ax + by + c = 0):

2x + 3y – 7 = 0

2ax + (a + b)y – 28 = 0

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

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

3. Compute `c_1/c_2 = (-7)/(-28)`

= `7/28`

= `1/4`

4. Set the other ratios equal to `1/4`:

`2/(2a) = 1/4`

⇒ `1/a = 1/4`

⇒ a = 4 

`3/(a + b) = 1/4` 

⇒ a + b = 12

⇒ With a = 4

⇒ b = 8