Questions: Solve the following system of equations. -8x + 3y = 9 10x - 3y = -15

Transcript text: Solve the following system of equations. \[ \begin{aligned} -8 x+3 y & =9 \\ 10 x-3 y & =-15 \end{aligned} \]

Solution

Solution Steps

To solve the system of linear equations, we can use matrix operations or substitution/elimination methods. Here, we will use the elimination method to eliminate one of the variables and solve for the other.

Step 1: Write the System of Equations

Given the system of equations: \[ \begin{aligned} -8x + 3y &= 9 \\ 10x - 3y &= -15 \end{aligned} \]

Step 2: Set Up the Matrix Equation

We can represent the system of equations in matrix form as: \[ A \mathbf{x} = \mathbf{B} \] where \[ A = \begin{bmatrix} -8 & 3 \\ 10 & -3 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 9 \\ -15 \end{bmatrix} \]

Step 3: Solve the Matrix Equation

To find \(\mathbf{x}\), we solve the matrix equation: \[ \mathbf{x} = A^{-1} \mathbf{B} \] Using the given output, the solution is: \[ \mathbf{x} = \begin{bmatrix} -3 \\ -5 \end{bmatrix} \]

Step 4: Interpret the Solution

The solution to the system of equations is: \[ x = -3, \quad y = -5 \]