Problem 45 - Precalculus: Mathematics for Calculus

Question

Use Cramer's Rule to solve the system. $$\left\\{\begin{array}{l} 0.4 x+1.2 y=0.4 \\ 1.2 x+1.6 y=3.2 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = -1\).

Step 1: Set Up the Coefficient Matrix

Identify the coefficients of the variables \(x\) and \(y\) from the system.The coefficient matrix \(A\) from the given system is:\[A = \begin{pmatrix} 0.4 & 1.2 \ 1.2 & 1.6 \end{pmatrix}\]This matrix contains all coefficients of \(x\) and \(y\) from the two equations.

Step 2: Calculate the Determinant of A

Calculate the determinant of the matrix \(A\), noted as \(\det(A)\).The determinant for a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is calculated as:\[\det(A) = ad - bc\]Apply this formula:\[\det(A) = (0.4)(1.6) - (1.2)(1.2) = 0.64 - 1.44 = -0.8\]

Step 3: Set Up the Matrices for Cramer's Rule

Cramer's Rule involves creating matrices \(A_x\) and \(A_y\), where each matrix is formed by replacing a specific column in \(A\) with the constant terms from the right-hand side of the equations.For \(A_x\): Replace the first column of \(A\) with the constants:\[A_x = \begin{pmatrix} 0.4 & 1.2 \ 3.2 & 1.6 \end{pmatrix}\]For \(A_y\): Replace the second column of \(A\) with the constants:\[A_y = \begin{pmatrix} 0.4 & 0.4 \ 1.2 & 3.2 \end{pmatrix}\]

Step 4: Calculate the Determinants of A_x and A_y

Calculate the determinants of \(A_x\) and \(A_y\), noted as \(\det(A_x)\) and \(\det(A_y)\) respectively.For \(A_x\):\[\det(A_x) = (0.4)(1.6) - (3.2)(1.2) = 0.64 - 3.84 = -3.2\]For \(A_y\):\[\det(A_y) = (0.4)(3.2) - (0.4)(1.2) = 1.28 - 0.48 = 0.8\]

Step 5: Apply Cramer's Rule to Find x and y

Cramer's Rule gives the solutions for \(x\) and \(y\):\[x = \frac{\det(A_x)}{\det(A)}\] and \[y = \frac{\det(A_y)}{\det(A)}\]Calculate \(x\):\[x = \frac{-3.2}{-0.8} = 4\]Calculate \(y\):\[y = \frac{0.8}{-0.8} = -1\]

Step 6: Substitution and Verification

Substitute the values of \(x\) and \(y\) back into the original equations to verify the solution. First equation:\[0.4(4) + 1.2(-1) = 1.6 - 1.2 = 0.4\] (Correct)Second equation:\[1.2(4) + 1.6(-1) = 4.8 - 1.6 = 3.2\] (Correct)The solutions satisfy both equations, confirming correctness.

Key Concepts

System of Equations

A system of equations is simply a set of two or more equations that you need to solve together. In our exercise, we have two equations with two unknowns, \(x\) and \(y\). This is a linear system because the equations are linear. Solving a system means finding values for \(x\) and \(y\) that satisfy both equations simultaneously.

• Equation 1: \(0.4x + 1.2y = 0.4\)
• Equation 2: \(1.2x + 1.6y = 3.2\)

The goal is to look for a common solution that works for both. Techniques like substitution, elimination, or matrix methods such as Cramer's Rule are usually employed to find this solution. The key challenge is ensuring both equations are balanced at the same time with the values of \(x\) and \(y\).

Determinant

The determinant is a special number calculated from a square matrix. It is crucial when using Cramer’s Rule. For our system, the determinant of matrix \(A\) helps determine if a unique solution exists. To compute the determinant of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), use the formula:\[\det(A) = ad - bc\]Calculating the determinant provides insight into the system’s solvability. If the determinant \(eq 0\), a unique solution exists. Here, the determinant is \(-0.8\), indicating a definitive solution, allowing us to proceed further with Cramer’s Rule.

Matrix Algebra

Matrix algebra involves working with matrices, providing an efficient way to handle multiple equations at once. During Cramer's Rule, we used a coefficient matrix \(A\) representing the equation's coefficients. This matrix simplifies complex systems by condensing information:\[A = \begin{pmatrix} 0.4 & 1.2 \ 1.2 & 1.6 \end{pmatrix}\]Through substitutions, matrices \(A_x\) and \(A_y\) replace columns with constant terms:

• \(A_x\) replaces the first column with constant terms: \(\begin{pmatrix} 0.4 & 1.2 \ 3.2 & 1.6 \end{pmatrix}\)
• \(A_y\) replaces the second column with constant terms: \(\begin{pmatrix} 0.4 & 0.4 \ 1.2 & 3.2 \end{pmatrix}\)

Manipulating these matrices lets us solve for individual variables using determinants.

Solution Verification

Solution verification is the crucial final step in solving equations. Once potential solutions \(x\) and \(y\) were found using determinants, it is important to substitute them back into the original equations. This ensures the correctness of your solutions.For Example:

• Plug \(x = 4\) and \(y = -1\) into Equation 1: \(0.4(4) + 1.2(-1) = 0.4\).
• Plug \(x = 4\) and \(y = -1\) into Equation 2: \(1.2(4) + 1.6(-1) = 3.2\).

Both verifications hold true in their respective equations, confirming that \(x = 4\) and \(y = -1\) are indeed correct. Verification reassures that the solutions captured the essence of the problem accurately, providing confidence in the processes used.