Question

Question
Use the elimination method to solve the following system of equations for x, y, and z. What is the value of z?

2
x
+
y

z
=
0


4
x

2
y
+
z
=
4



x
+
y
=
1


(1 point)
z

=

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Answers

Answered by GPT-4o mini
To solve the system of equations using the elimination method, let's first write down the equations clearly:

1. \( 2x + y - z = 0 \) (Equation 1)
2. \( 4x - 2y + z = 4 \) (Equation 2)
3. \( -x + y = 1 \) (Equation 3)

We’ll eliminate \( z \) from the first two equations.

### Step 1: Solve for \( z \) in terms of \( x \) and \( y \)

From Equation 1, we can express \( z \):
\[
z = 2x + y
\]

### Step 2: Substitute \( z \) in Equation 2

Now we substitute \( z \) into Equation 2:
\[
4x - 2y + (2x + y) = 4
\]
Simplifying this, we have:
\[
4x - 2y + 2x + y = 4
\]
\[
(4x + 2x) + (-2y + y) = 4
\]
\[
6x - y = 4
\]
We can rearrange this as:
\[
y = 6x - 4 \quad \text{(Equation 4)}
\]

### Step 3: Substitute Equation 4 into Equation 3

Now we'll go to Equation 3 and substitute for \( y \):
\[
-x + (6x - 4) = 1
\]
Simplifying this gives:
\[
-x + 6x - 4 = 1
\]
\[
5x - 4 = 1
\]
\[
5x = 5
\]
\[
x = 1
\]

### Step 4: Find \( y \) using Equation 4

Now substitute \( x = 1 \) back into Equation 4 to find \( y \):
\[
y = 6(1) - 4
\]
\[
y = 6 - 4
\]
\[
y = 2
\]

### Step 5: Find \( z \) using the expression for \( z \)

Finally, use the value of \( x \) and \( y \) in Equation 1 to find \( z \):
\[
z = 2(1) + 2
\]
\[
z = 2 + 2
\]
\[
z = 4
\]

### Final Answer

Thus, the value of \( z \) is:
\[
\boxed{4}
\]
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