5x-4y+2z=21, -1x-5y+6z=-24, -1x-4y+5z=-21

We have the following system of three linear equations:

\[
5x - 4y + 2z = 21
\]
\[
-x - 5y + 6z = -24
\]
\[
-x - 4y + 5z = -21
\]

To solve this system, we can use substitution, elimination, or matrix methods (like Gaussian elimination). Let's proceed with the elimination method to solve for \(x\), \(y\), and \(z\).

### Step 1: Eliminate \(x\) between the second and third equations.
The second and third equations are:

\[
-x - 5y + 6z = -24
\]
\[
-x - 4y + 5z = -21
\]

Subtract the third equation from the second to eliminate \(x\):

\[
(-x - 5y + 6z) - (-x - 4y + 5z) = -24 - (-21)
\]

Simplifying:

\[
-x + x - 5y + 4y + 6z - 5z = -24 + 21
\]
\[
-y + z = -3 \quad \text{(Equation 4)}
\]

### Step 2: Eliminate \(x\) between the first and second equations.
The first and second equations are:

\[
5x - 4y + 2z = 21
\]
\[
-x - 5y + 6z = -24
\]

Multiply the second equation by 5 to align the \(x\) terms:

\[
-5x - 25y + 30z = -120
\]

Now, add this equation to the first equation:

\[
(5x - 4y + 2z) + (-5x - 25y + 30z) = 21 + (-120)
\]

Simplifying:

\[
-29y + 32z = -99 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of two equations in two unknowns.
We now have two equations:

1. \(-y + z = -3\) (Equation 4)
2. \(-29y + 32z = -99\) (Equation 5)

From Equation 4, solve for \(z\) in terms of \(y\):

\[
z = y - 3
\]

Substitute this into Equation 5:

\[
-29y + 32(y - 3) = -99
\]

Simplify:

\[
-29y + 32y - 96 = -99
\]
\[
3y = -3
\]
\[
y = -1
\]

### Step 4: Solve for \(z\).
Substitute \(y = -1\) into \(z = y - 3\):

\[
z = -1 - 3 = -4
\]

### Step 5: Solve for \(x\).
Substitute \(y = -1\) and \(z = -4\) into one of the original equations, say the first one:

\[
5x - 4(-1) + 2(-4) = 21
\]
\[
5x + 4 - 8 = 21
\]
\[
5x - 4 = 21
\]
\[
5x = 25
\]
\[
x = 5
\]

### Final Solution:
The solution to the system of equations is:

\[
x = 5, \quad y = -1, \quad z = -4
\]

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:
1. How can you verify that this solution satisfies all three original equations?
2. How does Gaussian elimination compare with the method used here?
3. Can you express the system in matrix form and solve it using row reduction?
4. What happens if the system has no solution or infinitely many solutions?
5. How can this method be adapted to systems with more than three variables?

**Tip**: When solving systems of equations, it's often helpful to check your work by substituting the solution back into the original equations.

5x - 4y + 2z = 21, -x - 5y + 6z = -24, -x - 4y + 5z = -21

Learn how to solve the system of linear equations 5x - 4y + 2z = 21, -x - 5y + 6z = -24, -x - 4y + 5z = -21 using the elimination method. This problem covers essential linear algebra concepts and shows how to apply substitution and elimination methods step-by-step. Suitable for high school students in Grades 10-12.

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