Question

16) -6x-2y - ス=-17 5x+y-6z = 19 -4x-6y-6z = -20

Answer 1

To solve the system of equations:

$\begin{gathered} -6x-2y-z=-17 \\ 5x+y-6z=19 \\ -4x-6y-6z=-20 \end{gathered}$

We need to choose two sets of two equations and eliminate the same variable from those to get a 2 by 2 system that we can solve. If we choose the first and second equation and multiply the first one by -6 we get:

$\begin{gathered} 36x+12y+6z=102 \\ 5x+y-6z=19 \end{gathered}$

Now we add the equation to get:

$41x+13y=121$

Now we choose the second and third equation and change the sign of the second equations, then we get:

$\begin{gathered} 5x+y-6z=19 \\ 4x+6y+6z=20 \end{gathered}$

Adding them we have:

$9x+7y=39$

Now we have the system:

$\begin{gathered} 41x+13y=121 \\ 9x+7y=39 \end{gathered}$

To solve it we mutiply the first equation by 7 and the second one by -13, then we have:

$\begin{gathered} 287x+91y=847 \\ -117x-91y=-507 \end{gathered}$

Adding this equation we have:

$\begin{gathered} 170x=340 \\ x=\frac{340}{170} \\ x=2 \end{gathered}$

Now that we have the value of x we plug it in the second equation for the 2 by 2 system to get y:

$\begin{gathered} 9(2)+7y=39 \\ 18+7y=39 \\ 7y=39-18 \\ 7y=21 \\ y=\frac{21}{7} \\ y=3 \end{gathered}$

Finally to find z we plug the value of x and y in the first equation of the original 3 by 3 system, then:

$\begin{gathered} -6(2)-2(3)-z=-17 \\ -12-6-z=-17 \\ z=17-12-6 \\ z=-1 \end{gathered}$

Therefore the solution of the system is:

$\begin{gathered} x=2 \\ y=3 \\ z=-1 \end{gathered}$