Problem 34 - Algebra and Trigonometry

Question

\(29-44\) 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

x = \(\frac{2}{5}\), y = -1.

Step 1: Identify the coefficients

The given system of linear equations is \(10x - 17y = 21\) and \(20x - 31y = 39\). Identify the coefficients: \(a_1 = 10\), \(b_1 = -17\), \(c_1 = 21\); \(a_2 = 20\), \(b_2 = -31\), \(c_2 = 39\).

Step 2: Calculate the determinant of the coefficient matrix

The determinant \(D\) of the coefficient matrix is calculated as: \[D = \begin{vmatrix} 10 & -17 \ 20 & -31 \end{vmatrix} = (10)(-31) - (-17)(20)\].

Step 3: Simplify the determinant

Calculate the components: \((10)(-31) = -310\) and \((-17)(20) = -340\). Therefore, the determinant \(D = -310 + 340 = 30\).

Step 4: Calculate the determinant for x (Dₓ)

Replace the x-coefficients with the constants to find \(D_{x}\):\[D_{x} = \begin{vmatrix} 21 & -17 \ 39 & -31 \end{vmatrix} = (21)(-31) - (-17)(39)\].

Step 5: Simplify Dₓ

Calculate the components: \((21)(-31) = -651\) and \((-17)(39) = -663\). Therefore, \(D_{x} = -651 + 663 = 12\).

Step 6: Calculate the determinant for y (Dᵧ)

Replace the y-coefficients with the constants to find \(D_{y}\):\[D_{y} = \begin{vmatrix} 10 & 21 \ 20 & 39 \end{vmatrix} = (10)(39) - (21)(20)\].

Step 7: Simplify Dᵧ

Calculate the components: \((10)(39) = 390\) and \((21)(20) = 420\). Therefore, \(D_{y} = 390 - 420 = -30\).

Step 8: Solve for x and y

Using Cramer's Rule, x and y can be calculated as follows:\[ x = \frac{D_{x}}{D} = \frac{12}{30} = \frac{2}{5} \]\[ y = \frac{D_{y}}{D} = \frac{-30}{30} = -1 \]

Step 9: Verify the solution

Substitute \(x = \frac{2}{5}\) and \(y = -1\) back into the original equations to verify:For \(10x - 17y = 21\): \(10\left(\frac{2}{5}\right) + 17 = 21\). For \(20x - 31y = 39\): \(20\left(\frac{2}{5}\right) + 31 = 39\). Both checks confirm the solution.

Key Concepts

Determinant

A determinant is a special number that can be computed from a square matrix. In the context of solving a system of linear equations using Cramer's Rule, the determinant plays a crucial role. It helps us evaluate whether a unique solution exists.
For a 2x2 matrix:

• The determinant is calculated using the formula: \(D = ad - bc\), where \(a, b, c,\) and \(d\) are the elements of the matrix.
• If the determinant is zero, the system does not have a unique solution.
• Otherwise, the determinant provides information that leads us to the solution using Cramer’s Rule.

In our exercise, calculating the determinant of the coefficient matrix was the first step. It helped us confirm that a unique solution could be determined, as the determinant \(D = 30\) was not zero. Understanding how determinants function allows students to apply Cramer's Rule effectively.

System of Linear Equations

A system of linear equations involves finding values for variables that satisfy all equations within the system. Each equation in the system represents a line in a 2D space. In our example, the system given was:

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

When solving such systems, we often look for a single point that lies on both lines, indicating a solution.
In our problem, we used Cramer's Rule to find this solution. Cramer’s Rule can only be applied if:

• The system is square, meaning the number of equations matches the number of variables.
• The determinant of the coefficient matrix is non-zero to ensure a unique solution.

By applying Cramer's Rule, we found the intersection point to be \(x = \frac{2}{5}\) and \(y = -1\). This point satisfies both equations, providing the solution to the system.

Matrix Algebra

Matrix algebra is a vital mathematical tool for handling systems of equations. In matrix form, our system of equations is represented by aligning coefficients into a matrix:The coefficients from our system:

• Matrix \(A = \begin{pmatrix} 10 & -17 \ 20 & -31 \end{pmatrix}\)
• Constant matrix \(C = \begin{pmatrix} 21 \ 39 \end{pmatrix}\)

Matrix algebra uses determinants to solve linear equations efficiently by simplifying the computation process.

• The determinant of the coefficient matrix helps check the existence of a unique solution.
• Replacements of columns in the matrix are done to solve for variables \(x\) and \(y\), using Cramer's Rule.

Understanding matrix algebra empowers students to approach linear equations systematically, providing a straightforward path to finding solutions, as demonstrated in the step-by-step solution.