Solve the system of equations(If the system is dependent, enter a general solution in terms of c):

Call the three equations A, B and C. Note that if we double C we get the same constant 6 as B. So we can write B=2C. That is, 70x+12y+4z=29x+5y+3z. So 41x+7y+z=0. Similarly, note that A=-C. That is, 6x+y-z=-35x-6y-2z. So 41x+7y+z=0. And B=-2A: 29x+5y+3z=-12x-2y+2z, or 41x+7y+z=0. So, because we've involved all three equations and discovered a common equation, there's dependence between the three equations and therefore no unique solution, but we can take one of the assumed variables and consider it as we would a constant. Let's call that constant c. The three equations can now be rewritten 6x+y=c-3, 29x+5y=6-3c, 35x+6y=3-2c. We only need two of these to find x and y. Take the first two and multiply the first by 5 and subtract the second: 30x+5y=5c-15 subtract 29x+5y=6-3c and we get x=8c-21. Substitute this in the first equation and we get 6(8c-21)+y=c-3, from which y=123-47c. We can check the answer by putting in c=0, for example, and checking out the three original equations after substituting for x and y.

by Top Rated User (982k points)