To solve the given system of equations:
[tex]\[
\begin{cases}
4x - 5y + z = 9 \\
6x + 8y - z = 27 \\
3x - 2y + 5z = 40
\end{cases}
\][/tex]
we need to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations simultaneously.
1. Equation Setup:
The system of equations is already given in a clear format:
[tex]\[
4x - 5y + z = 9 \quad \text{(Eq1)}
\][/tex]
[tex]\[
6x + 8y - z = 27 \quad \text{(Eq2)}
\][/tex]
[tex]\[
3x - 2y + 5z = 40 \quad \text{(Eq3)}
\][/tex]
2. Solving the System:
We proceed by using substitution or elimination methods to solve this system.
A step-by-step method might involve algebraic manipulation such as adding and subtracting equations to eliminate variables. Here, to ensure we keep the process systematic and clear, we'll generally discuss the idea:
- Combine equations to eliminate [tex]\( z \)[/tex].
- Solve the resulting two equations for [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- Substitute [tex]\( x \)[/tex] and [tex]\( y \)[/tex] back into one of the original equations to solve for [tex]\( z \)[/tex].
3. Result:
After performing the necessary algebraic steps (which involves eliminating variables step-by-step and solving the reduced system), we find that the solution to the system is:
[tex]\[
x = 3, \quad y = 2, \quad z = 7.
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
(x, y, z) = (3, 2, 7)
\][/tex]
These values for [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] satisfy all three original equations simultaneously.
