Problem 22 - Algebra and Trigonometry

Question

Use Cramer’s Rule to solve each system. $$\left\\{\begin{aligned}y &=-4 x+2 \\\2 x &=3 y+8\end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations using Cramer's Rule is \(x = -1/7, y = 18/7\).

Step 1: Arrange The System of Equations and Label

We first arrange the system of equations in standard form and label the coefficients as \(a\), \(b\), and \(c\):\n\(4x + y = 2, -2x + 3y = 8\) where \(a = 4, b = 1, c = 2\) for the first equation and \(a = -2, b = 3, c = 8\) for the second equation.

Step 2: Form The Coefficient Matrix And Calculate Its Determinant

We form the coefficient matrix \(A = [[4, 1], [-2, 3]]\) and calculate its determinant to be \(D = |A| = (4*3) - (1*-2) = 14\).

Step 3: Form The Cramer's Matrices \(Dx\) and \(Dy\) And Calculate Their Determinants

We form two new matrices \(Dx\) and \(Dy\) by replacing the x and y coefficient columns in matrix A with the numbers on the right hand side of the equations. We then calculate their determinants: \(Dx = |[[2, 1], [8, 3]]| = (6 - 8) = -2\), \(Dy = |[[4, 2], [-2, 8]]| = (32 - (-4)) = 36\).

Step 4: Use Cramer's Rule to Solve for \(x\) and \(y\)

We use Cramer's Rule to find the solution: \(x = Dx / D = -2/14 = -1/7\), \(y = Dy / D = 36/14 = 18/7\).

Key Concepts

System of Linear Equations

Understanding the basics of a system of linear equations is pivotal when learning algebra. Essentially, this system consists of two or more linear equations with the same set of variables. Our goal is to find a common solution to these equations, which represents a point where the lines intersect if we were to graph them.
In the exercise provided, we have two such equations involving the variables 'x' and 'y.' To visualize, if you were to plot both equations on a graph, the solution would be where their lines would cross. Cramer's Rule offers a way to find this intersection without graphing, by working with the determinants of matrices, which are related to the coefficients of the variables in the equations.
While Cramer's Rule works only for systems where the number of equations and variables are the same, its utility in quickly finding solutions makes it an essential tool in linear algebra. For instance, in the given exercise, we use Cramer's Rule to solve for two variables, 'x' and 'y,' making it perfectly suitable.

Determinant of a Matrix

The determinant of a matrix, often symbolized as 'det(A)' or '|A|', is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible or not (a non-zero determinant implies the matrix can be inverted). The computation of a determinant involves a specific arithmetic manipulation of the matrix elements.
For a 2x2 matrix, the determinant is calculated as the difference between the products of its diagonals. Specifically, for matrix A with elements

• a in the top left,
• b in the top right,
• c in the bottom left, and
• d in the bottom right,

the determinant is ad - bc. In our exercise example, the determinant of the coefficient matrix A is calculated to be 14, which means that the system of linear equations represented by this matrix has a unique solution, since the determinant is not zero.

Coefficient Matrix

A coefficient matrix is a square matrix that contains only the coefficients of the variables from a system of linear equations. It is a powerful abstraction in linear algebra, as it allows for solving systems of equations using matrix operations.
In the exercise, the coefficient matrix A is constructed using the coefficients of 'x' and 'y' from the system of linear equations. The matrix A looks like this:

• The first row consists of the coefficients from the first equation of the system, which would represent the 'x' and 'y' coefficients, respectively.
• The second row contains the coefficients from the second equation.

Understanding the coefficient matrix is crucial for methods like Cramer's Rule because it leads directly to the calculation of matrix determinants necessary to find the values of the variables. When using Cramer's Rule, we manipulate this matrix to create what we call the Cramer's matrices (Dx and Dy in our exercise), which are essential in solving for each variable individually.