Question

please help me with this system of equations problem before I go insane the preferable solving methods would be through substitution or elimination

Answer 1

Answer:

x=8, y=0, z=-3

Step-by-step explanation:

Preferable is substitution because there is less room for mistakes but its long so here goes bare with me on this one :D

Name your equations

4x+2y-5z=47 (first eq)

x-2y+6z=-10 (second eq)

9x-7y-z=75 (third eq)

Lets take the easiest eq to isolate one variable lets take 2nd eq

$x-2y+6z=-10\\x=2y-6z-10$

now this becomes our fourth eq

now again take the easiest eq to isolate another variable lets take the 3rd eq

$9x-7y-z=75\\9x-7y-75=z$

now this becomes our fifth eq

and now plug in our fifth eq in the fourth eq because we now have z in terms of x and y

$x=2y-6z-10\\x=2y-6(9x-7y-75)-10\\x=2y-54x+42y+450-10\\x+54x=2y+42y+450-10\\55x=44y+440\\\\x=\frac{44y+440}{55}$

and now we got our sixth equation which is x only in terms of y.

now we plug in our sixth equation in our fifth eq to get z in terms of y only

$z=9x-7y-75\\\\z=9(\frac{44y+440}{55}) -7y-75\\z=\frac{396y+3960}{55} -7y-75\\z=\frac{396y+3960-385y-4125}{55} \\z=\frac{11y-165}{55}$

now we have our seventh eq and x &z in terms of just one variable y , now we put equation of z and x which is our sixth eq and seventh eq in our first eq.

$4x+2y-5z=47\\\\4(\frac{44y+440}{55})+2y-5(\frac{11y-165}{55}) =47\\\\\frac{176y+1760}{55} +2y-47=\frac{11y-165}{11} \\\\\frac{176y+1760}{55} -\frac{11y-165}{11} -47=-2y\\\\\frac{176y+1760}{55}-(y-15) -47=-2y\\\\\frac{176y+1760}{55} -y-32=-2y\\\\\\frac{176y+1760}{55}=-2y+y+32 \\\\176y+1760=55(-y+32)\\176y+1760=-55y+1760\\176y-55y=1760-1760\\121y=0\\y=0$

and now finally we got y=0 now we need the value of x and z, put the value of y=0 in our seventh eq (z equation)

$z=\frac{11y-165}{55} \\\\z=\frac{11(0)-165}{55} \\\\z=\frac{-165}{55}\\\\z=-3$

and now we put the value of y=0 in our sixth eq (x equation)

$x=\frac{44y+440}{55} \\\\x=\frac{44(0)+440}{55} \\\\x=\frac{440}{55} \\\\x=8$

and now we got all the three values and hence the solution of the system of equation is

x=8, y=0, z=-3