If (x, y) = (3, 4) is a solution of the simultaneous equations.[tex]ax \: + \: by \: = 4 \\ bx \: + \: ay \: = 8[/tex]find the value of a and b.​

The first thing you want to do is plug in x and y into both equations:

a(3) + b(4) = 4

b(3) + a(4) = 8

rearrange to line up a’s and b’s

3a + 4b = 4

4a + 3b = 8

now you want to choose a or b and multiply each equation by a number to make them have the same amount of a’s or b’s.

4(3a + 4b = 4) = 12a + 16b = 16

3(4a + 3b = 8) = 12a + 9b = 24


Now we subtract the bottom equation from the top and solve for b:

12a + 16b - (12a + 9b) = 16 - 24

7b = -8

b = -8/7


Now we plug back in for b to one of the original equations:

3a + 4(-8/7) = 4

3a + (-32/7) = 4

3a - (32/7) = 4

3a = 4 + (32/7)

3a = (28/7) + (32/7)

3a = 60/7

a = (60/7)/3 = 20/7.


Finally, plug a and b in together to double check using the second equation.

4a + 3b = 8

4(20/7) + 3(-8/7) = ?

(80/7) - (24/7) = ?

56/7 = 8.