Questions: 30x + 20y + 50z = 420 10x + 21y + 70z = 220

Transcript text: $\begin{array}{l}30 x+20 y+50 z=420 \\ 10 x+21 y+70 z=220\end{array}$

Solution

Solution Steps

To solve this system of linear equations, we can use matrix operations. We will represent the system in matrix form $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix. We can then use numpy's linear algebra solver to find the values of $x$, $y$, and $z$.

Step 1: Represent the System of Equations in Matrix Form

The given system of equations is: \[ \begin{align_} 30x + 20y + 50z &= 420 \\ 10x + 21y + 70z &= 220 \end{align_} \] We can represent this system in matrix form as $AX = B$, where: \[ A = \begin{bmatrix} 30 & 20 & 50 \\ 10 & 21 & 70 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 420 \\ 220 \end{bmatrix} \]

Step 2: Solve for the Variable Matrix $X$

To find the values of $x$, $y$, and $z$, we solve the matrix equation $AX = B$. The solution to this equation is given by: \[ X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 10.9299714 \\ 2.6058450 \\ 0.7996792 \end{bmatrix} \]

Step 3: Interpret the Solution

The solution to the system of equations is: \[ x \approx 10.93, \quad y \approx 2.606, \quad z \approx 0.7997 \]