Questions: Use the ALEKS graphing calculator to solve the system of equations. 0.45 x - y = -3.1 0.9 y = 3.6 x + 18 Round to the nearest hundredth. (x, y) = (?, ?)

Transcript text: Use the ALEKS graphing calculator to solve the system of equations. \[ \begin{array}{l} 0.45 x-y=-3.1 \\ 0.9 y=3.6 x+18 \end{array} \] Round to the nearest hundredth. \[ \left.(x, y)=\left(\prod\right], \square\right) \]

Solution

Solution Steps

To solve the system of equations, we can use a method such as substitution or elimination. However, a more straightforward approach is to use Python's numerical libraries to find the intersection point of the two lines represented by the equations. We will use NumPy to set up the equations in matrix form and solve for the variables \(x\) and \(y\).

Step 1: Set Up the System of Equations

We are given the system of equations: \[ \begin{align_} 0.45x - y &= -3.1 \\ 0.9y &= 3.6x + 18 \end{align_} \]

Step 2: Convert the Second Equation

Rearrange the second equation to match the standard form \(ax + by = c\): \[ 3.6x - 0.9y = -18 \]

Step 3: Express in Matrix Form

The system of equations can be expressed in matrix form as: \[ \begin{bmatrix} 0.45 & -1 \\ 3.6 & -0.9 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

\begin{bmatrix} -3.1 \\ -18 \end{bmatrix} \]

Step 4: Solve the System

Using matrix operations, we solve for \(x\) and \(y\): \[ \begin{bmatrix} x \\ y \end{bmatrix}

\begin{bmatrix} -4.76 \\ 0.96 \end{bmatrix} \]