Don't let the decimals frighten you! Call the equations A, B and C. Eliminate y between B and C by multiplying B by 5 and adding to C: 2.5x-y+2z=3.5 added to -1.2x+y-0.72z=0.9 comes to 1.3x+1.28z=4.4 Eliminate y between A and B: 2A+3B: 0.4x+0.6y+2.2z=3.2 plus 1.5x-0.6y+1.2z=2.1 comes to 1.9x+3.4z=5.3. Now we have two equations with the same two variables, x and z. Unfortunately, the numbers are becoming cumbersome to deal with, which suggest that the original question may not be correctly transcribed. This will be revealed as we push on. We need to multiply the first of these 2-variable equations by 1.9 and the second by 1.3 to eliminate x: 2.47x+2.432z=8.36 and 2.47x+4.42z=6.89. Subtracting the former from the latter we get 1.988z=-1.47. Clearly z is an awkward number and can't be expressed exactly in decimals. So we need to use fractions and z=-105/142. From this we can find x by substituting this value for z in 1.9x+3.4z=5.3. So x=292/71. We go back to one of the original equations to find y.
Let's pick A. 0.3y=1.6-1.1z-0.2x, from which y=753/142. We need to substitute these three values in B and C to check the integrity of the solution. Let's pick B first. 0.5*292/71-0.2*753/142+0.4(-105/142)=0.7. That's OK. Now C. -1.2*292/71+753/142-0.72(-105/142)=0.9. That's OK, too, so the solution is x=4.113, y=5.303, z=-0.739 to 3 places of decimals.
By the way, A checks out, too.
If you're wondering how to convert decimals into fractions, all you have to remember is to divide an integer by the appropriate number of powers of 10. So 0.72 is 72/100=18/25, for example. When you convert fractions into decimals you may get a recurring decimal.