Question

How would I solve this system of equation without using a maxtrix?
-1=A+C
7=-6A+B+D
-9=15A-6B+9C
24=15B+9D
I tried 6 (7=-6A+B+D) + 24=15B+9D = -39=-54A+6B
But then I am stuck

Answer 1

The quickest, most efficient method is Gaussian elimination. Take your variables in order, one at a time, going from A to B to C to D. So, start with A and a particular equation, take equation #1. Multiply equation #1 by a suitable number and add that equation (left to left and right to right) to each of the other equations (equation #2 and #3) to eliminate A from #2 and #3. For example, keep equation #1 as is, but for calculating, multiply it by 6 on both sides and add it to equation #2. The "a" will drop out of equation#2. Similar for the handling of equation #3. Multiply equation #1 by -15 and add it to equation #3. The "a" will drop out of equation #3. At this point, you will have an original equation #1 and #4 (because no "A" in it) and an altered #2 and #3. Repeat with the other variables and always picking an "unused" equation. Equation #1 is the only "used" equation at this time.

These instructions are long, but this is the quickest way, believe me. Once you get the methodology, you can always work these problems.