91Ó°ÊÓ

A system of equations is a collection of two or more equations with a set of variables. In this exercise, we are given a system involving two equations: \(x + 2y = 3\) and \(3x + 4y = 3\). These equations share the same variables, \(x\) and \(y\), and our goal is to find values for these variables that satisfy both equations simultaneously.

Solving a system of equations can be approached in several ways, such as substitution, elimination, or using matrices. When represented in matrix form, we can leverage matrix operations like inversion to efficiently find a solution. This is particularly useful for systems with multiple equations and variables. In this example, the matrix approach allows us to find the values of \(x\) and \(y\) by expressing the system as a product of matrices.

Matrix multiplication is a method of multiplying two matrices by each other. When solving systems of equations, this operation is essential because it enables us to manipulate and solve matrix equations.

Matrix multiplication involves taking the dot product of rows and columns. For this exercise, we multiply the inverse of the coefficient matrix \(\begin{pmatrix} -4 & 2 \ 3 & -1 \end{pmatrix}\) with the result matrix \(\begin{pmatrix} 3 \ 3 \end{pmatrix}\) to find the values of \(x\) and \(y\).

Here's a quick rundown of steps in matrix multiplication:

• Multiply the elements of the rows of the first matrix by the corresponding elements of the columns of the second matrix.
• Add the products together to form a single element in the resulting matrix.

In our example, using these rules helps us determine that \(x = -6\) and \(y = 9\).

The inverse of a matrix is analogous to dividing by a number in regular arithmetic. Specifically, for a matrix \(A\), its inverse \(A^{-1}\) is such that \(A \cdot A^{-1} = I\), where \(I\) represents the identity matrix.

Finding the inverse is crucial when solving systems of equations via matrix methods. When we have an equation in the form \(Ax = b\), multiplying both sides by \(A^{-1}\) helps isolate \(x\), thus allowing us to find the solutions of the system.

For a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), its inverse is given by \(\frac{1}{ad-bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}\), provided \(ad-bc eq 0\). In our exercise, this formula helps compute the inverse \(\begin{pmatrix} -4 & 2 \ 3 & -1 \end{pmatrix}\) of the coefficient matrix, which is used to find \(x = -6\) and \(y = 9\).

Understanding the inverse is key to solving matrix equations efficiently and effectively.