Problem 38 - 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 to the system is \(x = \frac{2}{5}\) and \(y = -1\).

Step 1: Write the System of Equations in Matrix Form

The given system of equations is: \[\begin{align*}10x - 17y &= 21 \20x - 31y &= 39\end{align*}\]This can be represented in matrix form as \(AX = B\) where:- \(A = \begin{bmatrix} 10 & -17 \ 20 & -31 \end{bmatrix}\)- \(X = \begin{bmatrix} x \ y \end{bmatrix}\)- \(B = \begin{bmatrix} 21 \ 39 \end{bmatrix}\).

Step 2: Calculate the Determinant of Matrix A

The determinant \(\Delta\) of matrix \(A\) is calculated as:\[\Delta = \begin{vmatrix} 10 & -17 \ 20 & -31 \end{vmatrix}\]Using the formula for a 2x2 determinant:\[\Delta = (10)(-31) - (20)(-17) = -310 + 340 = 30\]

Step 3: Formulate Matrices A1 and A2 for Cramer's Rule

To solve for \(x\) and \(y\), we use matrices \(A_1\) and \(A_2\):- Replace the first column of \(A\) with \(B\) to form \(A_1\):\[A_1 = \begin{bmatrix} 21 & -17 \ 39 & -31 \end{bmatrix}\]- Replace the second column of \(A\) with \(B\) to form \(A_2\):\[A_2 = \begin{bmatrix} 10 & 21 \ 20 & 39 \end{bmatrix}\]

Step 4: Calculate Determinants for A1 and A2

Find the determinants \(\Delta_1\) and \(\Delta_2\):- For \(A_1\):\[\Delta_1 = \begin{vmatrix} 21 & -17 \ 39 & -31 \end{vmatrix} = (21)(-31) - (39)(-17) = -651 + 663 = 12\]- For \(A_2\):\[\Delta_2 = \begin{vmatrix} 10 & 21 \ 20 & 39 \end{vmatrix} = (10)(39) - (20)(21) = 390 - 420 = -30\]

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

Applying Cramer's Rule:- \(x = \frac{\Delta_1}{\Delta} = \frac{12}{30} = \frac{2}{5}\)- \(y = \frac{\Delta_2}{\Delta} = \frac{-30}{30} = -1\)Thus, the solution is \(x = \frac{2}{5}\) and \(y = -1\).

Key Concepts

Determinant

The concept of a determinant is crucial when working with matrices, especially in solving systems of linear equations using methods like Cramer's Rule. The determinant is a special number that can be calculated from a square matrix.

For a 2x2 matrix, the formula for the determinant is relatively straightforward. Given a matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]the determinant, denoted as \( \Delta \), is calculated as:\[\Delta = (a \times d) - (b \times c).\]
This number is more than just a mathematical curiosity; it provides valuable information about a matrix. In the context of solving equations, if the determinant is zero, it implies that the system of equations doesn't have a unique solution. Otherwise, a non-zero determinant indicates that there is a unique solution.

Understanding determinants is critical, as they are foundational to various calculations in linear algebra and are used in multiple applications beyond just solving equations.

System of Equations

A system of equations involves dealing with two or more linear equations simultaneously. The challenge is to find a set of values for the variables involved that satisfy all the given equations simultaneously.

In the problem at hand, we are working with the following system of equations:

• \(10x - 17y = 21\)
• \(20x - 31y = 39\)

These equations form what's known as a **linear system**. This term suggests that the equations graphically represent straight lines, and solving the system amounts to finding the point where these lines intersect. In practice, this point provides the values of \(x\) and \(y\) that solve both equations.

Complex systems may require advanced methods like elimination, substitution, or matrix techniques such as Cramer's Rule, especially when dealing with larger systems.

Matrix Form

Writing a system of equations in matrix form is an efficient way to handle complex calculations. This method is not only organized but also simplifies the process of applying techniques like Cramer's Rule.

To transform the given system of equations into matrix form, we use:\[AX = B,\]where:

• \(A\) is the **coefficient matrix**, \(\begin{bmatrix} 10 & -17 \ 20 & -31 \end{bmatrix}\).
• \(X\) is the **variable matrix**, \(\begin{bmatrix} x \ y \end{bmatrix}\).
• \(B\) is the **constant matrix**, \(\begin{bmatrix} 21 \ 39 \end{bmatrix}\).

This representation in matrix form is a clean and concise way of organizing our information. It also sets the stage for using matrix operations and determinant calculations to systematically find solutions.

Solving Linear Equations

Solving linear equations involves finding the values of variables that satisfy each equation in a system. There are several methods to tackle such problems, but Cramer's Rule is a particularly structured approach.

Cramer's Rule uses the concepts of determinants and matrices to solve systems of linear equations that are exactly determined (i.e., have the same number of equations as unknowns).

• First, calculate the determinant of the coefficient matrix \(A\).
• Next, create modified matrices, \(A_1\) and \(A_2\), by replacing columns of \(A\) with the constant matrix \(B\).
• Find the determinants for these modified matrices.
• Finally, substitute into Cramer's Rule formulas: - \(x = \frac{\Delta_1}{\Delta}\) - \(y = \frac{\Delta_2}{\Delta}\)

This method is systematic and powerful, but it's essential to ensure that the determinant \(\Delta\) of \(A\) is not zero to apply Cramer's Rule. Otherwise, it implies that the solution is not unique, or there might be no solution at all. This structured approach helps verify that the solutions found are correct and unique.