Solving simultaneous equation using elimination

Equations with two equality signs:

Example: when it involves expectation with inequality signs, we need to further split it into two equations in order to solve it simultaneously. For instance. Find c and d in the expression below.

c + d + 6 = 2c - 5d -3 = 3c - 4d + 2.

Since we have two equality signs, we have to resolve it into simultaneous equations by pairing them and readjusting it to simultaneous equations as follows. 

 c + d + 6 =2c - 5d -3

Collect like terms:

c - 2c + d +5d = -3-6

-c +6d = -9  -----(I)

Also, 2c -5d -3 = 3c -4d + 2

CLT: 2c -3c -5d +4d = 2 +3

        -c -d = 5 -------(Ii)

 combining (I) and (Ii)

 -c + 6d = -9 ----(I)

  -c - d = 5 -------(Ii)

With the same sign and same coefficient, we eliminate c by subtracting equation (Ii) from equation (I).

We now have this after subtraction.

0 + 7d = -14

 7d = -14

d = -14/ 7

 d = -2

Substitute d in equation (I):

-c + 6d = -9

-c +6(-2) = -9

 -c -12 = -9

 CLT:

-c = -9 +.12

-c = 3

 c = -3

Therefore, c=-3, d=-2