Expert: Scott A Wilson - 2/1/2012

Question
Hi, I have a pair of equations that I have to solve simultaneously by using the substitution method.
The equations are:
(i) 7a - 5b + 6 = 0
(ii) a = 4b + 1
The way I've done it, I just substitute (ii) into (i) to give
7(4b + 1) - 5b + 6 = 0 and then solve like a regular equation.
I always end up with b = -13/19 and a = -33/19 but the answer is a = -29/23 and b = -13/23.
Where am I going wrong?
Thanks for your help,
Erin

Answer
The equation 7(4b + 1) - 5b + 6 = 0 is correct.

If we multiply the 7 by 4b we get 28b.
If we multiply the 7 by 1 we get 7.
This makes the equation into 28b + 7 - 5b + 6 = 0.

If we combine the b terms, we get (28-5)b, which is the same as 23b.
If we combine the constants, we get 7+6, which is 13.

So far, then, we have 23b + 13 = 0.
Subtracting 13 from both sides gives 23b = -13.
Dividing both sides by 23 gives b = -13/23.

Since a = 4b + 1, and it is known that we can say the 1 is really 23/23,
we have a = 4(-13/23) + 23/23 = -52/23 + 23/23 = (-52+23)/23 = -29/23.

Checking this against (i) gives 7(-29/23) - 5(-13/23) + 6 =
-203/23 + 65/23 + 138/23.

Since 138+65 is 203, the value comes out to be 0, and that is as it should be.