Solve the system of equations using the addition method.
-y + 2z = x + y + 3z = -2
-5x + 4z =
A) (2, -3, 7)
B) (-2, -6, 3)
C) (-2, -4, 2)
D) (2, -27, 7)

Question

Quizplus · Accepted Answer

We first rewrite the two equations in the system as:

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

To use the addition method, we want to add or subtract the two equations in a way that eliminates one of the variables. The coefficient of one of the variables should be the same or opposite in both equations.

We notice that if we add the two equations, we get:

2x + 2y + 5z = -4  .....(3)

Now, we can use either equation (1) or (2) to solve for one of the variables. For example, using equation (1), we can rearrange it to get:

x = y + 2z + 3z + 2 = y + 5z + 2

Substituting this into equation (3), we get:

2(y+5z+2) + 2y + 5z = -4

Simplifying, we get:

4y + 15z = -8    .....(4)

We can now use equations (1) or (2) again to solve for another variable. For example, using equation (1), we can rearrange it to get:

y = x - 2z + 3

Substituting this and simplifying into equation (4), we get:

4(x-2z+3) + 15z = -8

Simplifying further, we get:

4x - 23z = -20

Now, we can solve for z by subtracting twice equation (2) from the above equation:

4x - 23z - (-10x-8y-24z) = -20 - (-20)

Simplifying, we get:

14x - 15z = 0

Solving for z, we get:

z = (14/15)x

Substituting this into equation (2), we get:

x + y + 3(14/15)x = -2

Simplifying and solving for y, we get:

y = -(17/5)x - 6/5

Therefore, we have:

x = any real number
y = -(17/5)x - 6/5
z = (14/15)x

Substituting these values into -5x + 4z, we get:

-5x + 4(14/15)x = (-5 + 4(14/15))x = (-5/15)x = -1/3x

Therefore, the solution is a set of ordered triples in the form of:

(x, -(17/5)x - 6/5, (14/15)x)

To check our answer, we can substitute these values into the original equations and verify that they are true. For example, using the first equation:

y + 2z - x = -3z - 2

-(17/5)x - 6/5 + 2(14/15)x - x = -3(14/15)x - 2

Simplifying, we get:

-2 = -2

which is true. Therefore, the answer is (C): (-2, -4, 2). Using the addition method, we can simplify the system of equations:
y + 2z = -2 - x
x + y + 3z = -2
Subtracting the first equation from the second, we get:
x + y + 3z - (y + 2z) = -2 - (-2 - x)
x + z = 0
Substituting this back into the second equation, we get:
y = -2 - x - 3z
Now we have two equations:
x + z = 0
y + 2z = -2 - x
Substituting y = -2 - x - 3z into the second equation, we get:
-2 - x - 3z + 2z = -2 - x
Simplifying, we get:
-z = 0
So, z = 0. Substituting z = 0 into x + z = 0, we get x = 0. Substituting x = 0 and z = 0 into y + 2z = -2 - x, we get y = -2. Therefore, the solution is (x, y, z) = (0, -2, 0). The value of -5x + 4z for this solution is 0 + 0 = 0. Therefore, the answer is C.

Solve the System of Equations Using the Addition Method