Question

Solve the system using the elimination method.

$2x+y-z=9$

$-x+6y+2z=-17$

$5x+7y+z=4$

The solution is $\left(x,y,z\right)=$ (
,
,
).
solve easily and simple

Answer 1

To solve the system of equations using the elimination method, we have the following three equations:

1. \( 2x + y - z = 9 \) (Equation 1)
2. \( -x + 6y + 2z = -17 \) (Equation 2)
3. \( 5x + 7y + z = 4 \) (Equation 3)

**Step 1: Eliminate \(z\) from Equations 1 and 2.**

From Equation 1, we can express \(z\) in terms of \(x\) and \(y\):
\[
z = 2x + y - 9
\]

Substituting this expression for \(z\) into Equation 2:
\[
-x + 6y + 2(2x + y - 9) = -17
\]

Expanding:
\[
-x + 6y + 4x + 2y - 18 = -17
\]
Combining like terms:
\[
(4x - x) + (6y + 2y) - 18 = -17 \\
3x + 8y - 18 = -17
\]
Adding 18 to both sides:
\[
3x + 8y = 1 \quad \text{(Equation 4)}
\]

**Step 2: Eliminate \(z\) from Equations 1 and 3.**

Using \(z = 2x + y - 9\) again, substitute into Equation 3:
\[
5x + 7y + (2x + y - 9) = 4
\]
Simplifying:
\[
5x + 7y + 2x + y - 9 = 4
\]
Combining like terms:
\[
(5x + 2x) + (7y + y) - 9 = 4\\
7x + 8y - 9 = 4
\]

Adding 9 to both sides:
\[
7x + 8y = 13 \quad \text{(Equation 5)}
\]

**Step 3: Solve the new system of equations (Equations 4 and 5).**

We have now simplified the system to:
1. \( 3x + 8y = 1 \) (Equation 4)
2. \( 7x + 8y = 13 \) (Equation 5)

Subtract Equation 4 from Equation 5:
\[
(7x + 8y) - (3x + 8y) = 13 - 1
\]
This simplifies to:
\[
4x = 12
\]
Dividing by 4:
\[
x = 3
\]

**Step 4: Substitute \(x\) back into one of the equations to solve for \(y\).**

Using Equation 4 to find \(y\):
\[
3(3) + 8y = 1 \\
9 + 8y = 1
\]
Subtracting 9 from both sides:
\[
8y = 1 - 9 \\
8y = -8
\]
Dividing by 8:
\[
y = -1
\]

**Step 5: Substitute \(x\) and \(y\) back to find \(z\).**

Using the expression for \(z\):
\[
z = 2(3) + (-1) - 9 \\
z = 6 - 1 - 9 \\
z = -4
\]

**Final solution:**
The solution to the system of equations is:
\[
(x, y, z) = (3, -1, -4)
\]