Question

1. determine the solution to the system below. desmos matrix calculator

2x + 2y + 2z = 10
x - 3y + 4z = 18
x + y = -2z
(5, 4, 2)
(17, -7, -5)
(-5, 17, -7)
(-4, 14, -5)
clear all

Explanation:

Step1: Rewrite the system in standard form

The system of equations is:

$$\begin{cases} 2x + 2y+2z = 10\\ x - 3y+4z = 18\\ x + y+2z = 0 \end{cases}$$

Step2: Use elimination method

First, subtract the second - equation from the third - equation:
$(x + y+2z)-(x - 3y+4z)=0 - 18$
$x + y+2z - x + 3y-4z=-18$
$4y-2z=-18$ (Equation 4)
Multiply the second - equation by 2: $2x-6y + 8z = 36$
Subtract it from the first - equation: $(2x + 2y+2z)-(2x-6y + 8z)=10 - 36$
$2x + 2y+2z-2x + 6y-8z=-26$
$8y-6z=-26$ (Equation 5)

Step3: Solve the new system of two - equations

Multiply Equation 4 by 2: $8y-4z=-36$
Subtract Equation 5 from it: $(8y-4z)-(8y-6z)=-36+26$
$8y-4z - 8y + 6z=-10$
$2z=-10$
$z=-5$

Step4: Find the value of y

Substitute $z = - 5$ into Equation 4: $4y-2\times(-5)=-18$
$4y + 10=-18$
$4y=-18 - 10$
$4y=-28$
$y=-7$

Step5: Find the value of x

Substitute $y=-7$ and $z = - 5$ into the third original equation $x + y+2z = 0$
$x-7+2\times(-5)=0$
$x-7-10=0$
$x=17$

Answer:

$(17,-7,-5)$