Is there a solution set?

Question

X=7y-8

3x-21y=-24

Answer 1

yes, there is a solution set,

So the first step when seeing these types of problems is to rearrange this system of equations so that all of the variable terms are on the left and all of the numbers are on the right. So now this system reads:

x-7y =-8

3x-21y = -24

The next step is to isolate the x and y in separate equations so that you can solve for each variable separately with normal algebra. You have two tools to do this:

1) scaling, where you multiply both sides of an equation by a single number

2) row replacement, where you replace one row with the sum of itself and another row (or any scaled multiple of another row).

Normally this process is done with matrices, but I am not sure how familiar you are with them and this word processing window can't really write them out.

So applying these two tools: first scale the second equation by -1/3 so now the two equations are

x-7y=-8

-x+7y = 8

then replace the second equation with itself plus the first row:

x-7y = -8

0=0

Ok, so we were unable to separate the variables. When this happens, the system of equations is linearly dependent and as long as all equations are valid this equation will have an infinite set of solutions. Luckily, the equation: 0=0 is valid so all values of x and y that satisfy the top equation are in the solution set.

We can solve for y in terms of x

y = (x+8)/7

So the set of all solutions is [x, (x+8)/7] for any real number x