Question

algebra 2 - test
systems of 3 variables
show all work for credit use a pencil
solve the system of equations using the elimination method.

1. ( x - 2y + 3z = 12 )

( 2x - y - 2z = 5 )
( 2x + 2y - z = 4 )
(and)
( (x - 2y + 3z) + (2x + 2y - z) = 12 )
( 3x + 2z = 16 )
( 2(3x + 2z) = 2×16 )
( 6x + 4z = 32 )
( 3x + 2(2) = 16 )
( 3x + 4 = 16 )
( 3x = 12 )
( x = 4 )
( 2(4) - y - 2(2) = 5 )
( 8 - y - 4 = 5 )
( -y = 1 )
solve the system of equations using substitution.

1. ( x - 2y + z = -6 )

( x = -5z - 12 )
( -x + 6y + 4z = 3 )

Explanation:

Step1: Solve the first system (elimination)

We have the system:

$$
LATEXBLOCK0
$$

First, add equation (1) and (3) to eliminate \(y\):
\((x - 2y + 3z)+(2x + 2y - z)=12 + 4\)
\(3x + 2z = 16 \quad (4)\)
Multiply equation (2) by 2: \(4x - 2y - 4z = 10 \quad (5)\)
Add equation (5) and (3): \((4x - 2y - 4z)+(2x + 2y - z)=10 + 4\)
\(6x - 5z = 14 \quad (6)\)
Now, multiply equation (4) by 2: \(6x + 4z = 32 \quad (7)\)
Subtract equation (6) from (7): \((6x + 4z)-(6x - 5z)=32 - 14\)
\(9z = 18\)
\(z = 2\)
Substitute \(z = 2\) into (4): \(3x + 2(2)=16\)
\(3x + 4 = 16\)
\(3x = 12\)
\(x = 4\)
Substitute \(x = 4\) and \(z = 2\) into (2): \(2(4)-y - 2(2)=5\)
\(8 - y - 4 = 5\)
\(4 - y = 5\)
\(-y = 1\)
\(y = - 1\)
So the solution for the first system is \((x,y,z)=(4,-1,2)\)

Step2: Solve the second system (substitution)

We have the system:

$$
LATEXBLOCK1
$$

Substitute (2) into (1): \((-5z - 12)-2y + z=-6\)
\(-4z - 2y - 12=-6\)
\(-4z - 2y=6\)
\(2z + y=-3 \quad (4)\)
Substitute (2) into (3): \(-(-5z - 12)+6y + 4z = 3\)
\(5z + 12 + 6y + 4z = 3\)
\(9z + 6y=-9\)
\(3z + 2y=-3 \quad (5)\)
From (4): \(y=-3 - 2z\)
Substitute into (5): \(3z + 2(-3 - 2z)=-3\)
\(3z-6 - 4z=-3\)
\(-z - 6=-3\)
\(-z = 3\)
\(z=-3\)
Substitute \(z = - 3\) into (2): \(x=-5(-3)-12=15 - 12 = 3\)
Substitute \(x = 3\) and \(z = - 3\) into (1): \(3-2y-3=-6\)
\(-2y=-6\)
\(y = 3\)
So the solution for the second system is \((x,y,z)=(3,3,-3)\)

Answer:

First system solution: \((4, - 1, 2)\)
Second system solution: \((3, 3, - 3)\)