Solve the system of equations:
$\begin{array}{l}x+9 y+8 z=0 \\ 5 x+8 y+z=35 \\ x-4 y-z=17\end{array}$
We are given a system of three linear equations with three variables:
\[ \begin{cases} (1) \quad x + 9y + 8z = 0 \ (2) \quad 5x + 8y + z = 35 \ (3) \quad x - 4y - z = 17 \end{cases} \]
Our goal is to find the values of \(x, y,\) and \(z\) that satisfy all three equations simultaneously. We will use the method of substitution to reduce the system step-by-step.
From equation (3), we can isolate the variable \(z\):
\[z = x - 4y - 17\]
Now, we substitute this expression for \(z\) into equations (1) and (2) to create a new system with only two variables, \(x\) and \(y\).
Substitute into equation (1):
\[x + 9y + 8(x - 4y - 17) = 0\]
\[x + 9y + 8x - 32y - 136 = 0\]
\[9x - 23y = 136 \quad (\text{Equation } 4)\]
Substitute into equation (2):
\[5x + 8y + (x - 4y - 17) = 35\]
\[6x + 4y - 17 = 35\]
\[6x + 4y = 52\]
Dividing by 2, we get:
\[3x + 2y = 26 \quad (\text{Equation } 5)\]
We now have the system:
\[ \begin{cases} (4) \quad 9x - 23y = 136 \ (5) \quad 3x + 2y = 26 \end{cases} \]
From equation (5), express \(3x\) in terms of \(y\):
\[3x = 26 - 2y\]
Multiply by 3 to get \(9x\):
\[9x = 78 - 6y\]
Substitute this into equation (4):
\[(78 - 6y) - 23y = 136\]
\[78 - 29y = 136\]
\[-29y = 136 - 78\]
\[-29y = 58\]
\[y = -2\]
Now substitute \(y = -2\) back into equation (5) to solve for \(x\):
\[3x + 2(-2) = 26\]
\[3x - 4 = 26\]
\[3x = 30\]
\[x = 10\]
Finally, use the expression for \(z\) from step 2 to find \(z\):
\[z = x - 4y - 17\]
\[z = 10 - 4(-2) - 17\]
\[z = 10 + 8 - 17\]
\[z = 1\]
We found the solution to be \(x = 10, y = -2, z = 1\). Let us verify these values in the original equations:
(1) \(10 + 9(-2) + 8(1) = 10 - 18 + 8 = 0\) (Correct)
(2) \(5(10) + 8(-2) + 1 = 50 - 16 + 1 = 35\) (Correct)
(3) \(10 - 4(-2) - 1 = 10 + 8 - 1 = 17\) (Correct)
The solution is verified and complete. Thus, the set of solutions is \(x=10\), \(y=-2\), and \(z=1\). In coordinate form, this is \((x, y, z) = (10, -2, 1)\).
Systems of linear equations are widely used in economics and logistics to balance resources. For example, if a factory produces three types of widgets, each requiring different amounts of materials (like steel, plastic, and energy) and different labor hours, a system of equations can help determine exactly how many of each widget can be produced given a fixed budget and supply of materials. Solving these systems ensures that the production plan meets all constraints efficiently, preventing waste and maximizing output.