Problem 46 - Precalculus: Mathematics for Calculus

Question

Use Cramer's Rule to solve the system. $$\left\\{\begin{array}{l} 10 x-17 y=21 \\ 20 x-31 y=39 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{2}{5} \) and \( y = -1 \).

Step 1: Write the System in Matrix Form

First, express the equations as a matrix equation in the form of \( AX = B \). The coefficients of \( x \) and \( y \) become the matrix \( A \), while the constants form \( B \). Matrix \( A \) is:\[A = \begin{bmatrix} 10 & -17 \ 20 & -31 \end{bmatrix}\]Vector \( X \) is:\[X = \begin{bmatrix} x \ y \end{bmatrix}\]Vector \( B \) is:\[B = \begin{bmatrix} 21 \ 39 \end{bmatrix}\]

Step 2: Calculate the Determinant of A

The next step is to find the determinant of matrix \( A \), denoted as \( \text{det}(A) \). Use the formula for a 2x2 matrix: \[\text{det}(A) = ad - bc\]Substitute the values:\[\text{det}(A) = (10)(-31) - (20)(-17) = -310 + 340 = 30\]

Step 3: Calculate Determinant \( D_x \) by Replacing First Column

To find \( x \) using Cramer's Rule, replace the first column of \( A \) with \( B \) and calculate the determinant \( D_x \). Matrix with replaced column:\[\begin{bmatrix} 21 & -17 \ 39 & -31 \end{bmatrix}\]Calculate \( D_x \):\[D_x = (21)(-31) - (39)(-17) = -651 + 663 = 12\]

Step 4: Calculate Determinant \( D_y \) by Replacing Second Column

Now replace the second column of \( A \) with \( B \) and calculate the determinant \( D_y \) to find \( y \).Matrix with replaced column:\[\begin{bmatrix} 10 & 21 \ 20 & 39 \end{bmatrix}\]Calculate \( D_y \):\[D_y = (10)(39) - (20)(21) = 390 - 420 = -30\]

Step 5: Solve for x and y Using Cramer's Rule

Use the determinants to solve for \( x \) and \( y \) using Cramer's Rule:\[ x = \frac{D_x}{\text{det}(A)} = \frac{12}{30} = \frac{2}{5} \]\[ y = \frac{D_y}{\text{det}(A)} = \frac{-30}{30} = -1 \]

Step 6: Write the Solution

The solution to the system of equations using Cramer's Rule is:\[(x, y) = \left( \frac{2}{5}, -1 \right)\]

Key Concepts

Determinants

Determinants are a fundamental concept in matrix algebra that help determine specific properties of matrices, such as invertibility. For a 2x2 matrix, the determinant can be calculated using a simple formula:

• For a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( \text{det}(A) = ad - bc \).

In the context of solving systems of equations with Cramer's Rule, determinants are used to evaluate whether the system has a unique solution. If the determinant of the coefficient matrix (matrix \( A \)) is non-zero, the system has a unique solution. If the determinant is zero, the system may have no solutions or infinitely many solutions.
When using Cramer's Rule, you also replace columns of the matrix to find determinants like \( D_x \) and \( D_y \). These determinants further help in computing the values of the variables, confirming if they can be uniquely determined based on the given equations.

Systems of Equations

Systems of equations are sets of equations with multiple variables that you solve simultaneously. The goal is to find values for the variables that satisfy all the equations in the system.
For example, the system:\[\begin{cases} 10x - 17y = 21 \ 20x - 31y = 39 \end{cases}\]requires determining \( x \) and \( y \) such that both equations hold true at the same time.

• These systems can be solved through various methods, including graphing, substitution, elimination, and matrix methods like Cramer's Rule.
• Cramer's Rule is particularly useful for systems where the number of equations equals the number of variables, and the system can be represented by square matrices.

This approach leverages determinants to systematically solve for each variable, offering a direct computation method especially useful for small systems.

Matrix Algebra

Matrix algebra is a branch of mathematics involving sets of numbers arranged in rectangular grids or tables. These grids are called matrices, composed of rows and columns. Matrix operations include addition, multiplication, and finding inverses and determinants.
When solving systems of equations using matrix algebra, each equation is expressed in matrix form. For a system \( AX = B \), matrix \( A \) contains the coefficients of the variables, \( X \) is the column matrix of variables, and \( B \) is the column matrix of constants.

• In matrix algebra, operations follow specific rules. For instance, multiplying two matrices requires that the number of columns in the first matrix matches the number of rows in the second matrix.
• Cramer's Rule is an application of matrix algebra, focusing on using determinants to find solutions to linear systems. When matrix \( A \) is invertible, which occurs when \( \text{det}(A) eq 0 \), Cramer's Rule provides a direct way to find the unique solution to the system.