Question

Solve each set of equations by the method of finding the inverse of the coefficient matrix. $$ \left\\{\begin{array}{l} 2 x+3 y=-1 \\ 5 x+4 y=8 \end{array}\right. $$

Short answer

The solution is \( x = 4 \) and \( y = -3 \).

Step-by-step solution

Write the equation in matrix form

Rewrite the system of equations in the form of a matrix equation. This can be done by writing the coefficients of the variables as a matrix, the variables as another matrix, and the constants on the other side: \[\begin{pmatrix} 2 & 3 \ 5 & 4 \end{pmatrix}\begin{pmatrix} x \ y \end{pmatrix}=\begin{pmatrix} -1 \ 8 \end{pmatrix}\]

Find the inverse of the coefficient matrix

To find the solution, we need to find the inverse of the coefficient matrix. The inverse of a 2x2 matrix \[\begin{pmatrix} a & b \ c & d \end{pmatrix}\] is given by \[ \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]. For our matrix, \[ A = \begin{pmatrix} 2 & 3 \ 5 & 4 \end{pmatrix}, \] we need to find \[ \text{det}(A) = 2\cdot4 - 3\cdot5 = 8 - 15 = -7. \] Using this determinant, the inverse is \[ A^{-1} = \frac{1}{-7} \begin{pmatrix} 4 & -3 \ -5 & 2 \end{pmatrix} = \begin{pmatrix} -\frac{4}{7} & \frac{3}{7} \ \frac{5}{7} & -\frac{2}{7} \end{pmatrix} \].

Multiply the inverse matrix by the constants

Next, we multiply the inverse of the coefficient matrix by the constants matrix to find the values of x and y. We perform the multiplication as follows: \[\begin{pmatrix} -\frac{4}{7} & \frac{3}{7} \ \frac{5}{7} & -\frac{2}{7} \end{pmatrix}\begin{pmatrix} -1 \ 8 \end{pmatrix} = \begin{pmatrix} -\frac{4}{7} \cdot (-1) + \frac{3}{7} \cdot 8 \ \frac{5}{7} \cdot (-1) + (-\frac{2}{7}) \cdot 8 \end{pmatrix} \]Solving the multiplication:\[= \begin{pmatrix} \frac{4}{7} + \frac{24}{7} \ -\frac{5}{7} - \frac{16}{7} \end{pmatrix} = \begin{pmatrix} \frac{28}{7} \ -\frac{21}{7} \end{pmatrix} = \begin{pmatrix} 4 \ -3 \end{pmatrix} \]

State the solution

The coordinates \( x \) and \( y \) that solve the system of equations are found in the resulting matrix. Thus, the solution to the system is \( x = 4 \) and \( y = -3 \).

Key concepts

Solving Linear Equations

In algebra, solving linear equations is a fundamental skill. Linear equations take the form of Ax + By = C, where A, B, and C are constants, and x and y are the variables we need to solve for. In a system of linear equations, we have multiple such equations that need to be satisfied simultaneously. One effective method to solve these is by using matrices.

By representing the coefficients of the variables in a matrix form, we can apply different matrix operations to find the solution. For our given problem, the equations are:

• 2x + 3y = -1


• 5x + 4y = 8

These equations can be expressed in matrix form, which helps in applying methods like matrix inversion to solve for x and y.

Matrix Inversion

To solve the system using the inverse matrix method, we convert our system into a matrix equation of the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constants matrix. In our problem, A is:
\[\begin{pmatrix} 2 & 3 \ 5 & 4 \end{pmatrix}\]
The inverse of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is calculated as:
\[\frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}\]
First, we find the determinant:
\[det(A) = 2 \cdot 4 - 3 \cdot 5 = 8 - 15 = -7\]
Then, we determine the inverse matrix A⁻¹:
\[A^{-1} = \frac{1}{-7} \begin{pmatrix} 4 & -3 \ -5 & 2 \end{pmatrix} = \begin{pmatrix} -\frac{4}{7} & \frac{3}{7} \ \frac{5}{7} & -\frac{2}{7} \end{pmatrix}\]
By using the inverse matrix, we can multiply it by the constants matrix to get the solution.

Coefficient Matrix

The coefficient matrix plays a crucial role in solving systems of linear equations using the inverse matrix method. It consists of the coefficients from the linear equations. For our problem, the coefficient matrix A is:
\[\begin{pmatrix} 2 & 3 \ 5 & 4 \end{pmatrix}\]
This matrix captures the essence of how the variables in the linear equations relate to each other. By finding the inverse of this matrix, we can multiply it with the constants matrix to isolate the variables. The constants matrix B is:
\[\begin{pmatrix} -1 \ 8 \end{pmatrix}\]
The step-by-step inversion and multiplication provide us with the values of x and y that satisfy both equations. In the end, we find that the solution to our system is:
\[\begin{pmatrix} 4 \ -3 \end{pmatrix}\]
Thus, x = 4 and y = -3. Understanding the role of the coefficient matrix is key to mastering the inverse matrix method for solving linear equations.