An engineer worked for 6 days and his assistant worked for 7 days earning a combined salary of $4500.00. The next week the engineer worked 5 days and his assistant worked 3 days for a combined salary of $2900.00. What does each person earn per day?

Solution: 6E + 7A = $4500.00
5E + 3A = $2900.00

-3 * [6E +7A = $4500.00] = -18E -21A = -13,500
7 * [5E +3A = $2900.00] = 35E +21A = 20,300

17E = 6800

E = $400.00/day

What they did was eliminate the A term by multiplying the first equation by -3 and the second equation by +7, then adding the two equations together to get a single equation in E without any A terms. So you have this:

6E + 7A = $4500.00
5E + 3A = $2900.00

-3 * [6E +7A = $4500.00] = -18E -21A = -13,500
7 * [5E +3A = $2900.00] = 35E +21A = 20,300

Now add:

-18E -21A +35A +21A = -13500 + 20300

Notice how the A terms cancel out? You get:
17E = 6800
E = 400.

Thank you. I get what he did. My question is more specific. Is there a defined "rule of elimination" that applies here and is this process an "allowed" rule or a "must" rule. Must you change the sign or are you allowed to change the sign?

The goal is to multiply the two equations by factors that will allow you to either add or subtract one equation from the other and eliminate one of the unknowns. It really doesn't matter whether you add or subtract, or use a positive or negative factor, or which of the two unknowns you eliminate. It's all legal. So for example, consider:

A + 2B = 8
A + B = 5

From inspection you can see that subtracting the 2nd from the 1st will elimniate the A term:

A+2B - (A+B) = 8-5
B=3
Then back substitute to find A = 2.

But you could have multiplied the second by -2 and added:

A+2B + -2(A+B) = 8+-2(5)
-A = -2
A = 2.

Whether you need to add or subtract is dependent on the signs of the terms. In the example above I subtracted one from the other. But consider this example:

A + 2B = 8
3A - B = 3

Here you can see that multiplying the second equation by 2 and adding it to the first will eliminate B:

A + 2B + 2(3A - B)= 8 + 2(3)
7A = 14
A=2.

So the determination of whether to add or subtract, or to use a positive factor or a negative, is entirely up to you. Use whatever makes the problem easiest to work.