Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}5 x+2 y+z=15 \\\2 x-y+z=9 \\\4 x+3 y+2 z=13\end{array}$$

Step-by-Step Solution

Answer

The solution is \(x = 3\), \(y = -1\), \(z = 2\).

Step 1: Express the System in Matrix Form

The given system of equations is:\[\begin{align*} 5x + 2y + z &= 15 \ 2x - y + z &= 9 \ 4x + 3y + 2z &= 13 \end{align*}\] We can write this in matrix form as \(AX = B\), where:\[A = \begin{bmatrix} 5 & 2 & 1 \ 2 & -1 & 1 \ 4 & 3 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 15 \ 9 \ 13 \end{bmatrix}\]

Step 2: Calculate the Determinant of Matrix A

The determinant \(D\) of the matrix \(A\) is calculated as follows:\[D = \begin{vmatrix} 5 & 2 & 1 \ 2 & -1 & 1 \ 4 & 3 & 2 \end{vmatrix}\]Using the rule of Sarrus or expansion by minors, calculate:\[D = 5((-1)(2) - (1)(3)) - 2((2)(2) - (1)(4)) + 1((2)(3) - (-1)(4))\]\[D = 5(-2 - 3) - 2(4 - 4) + 1(6 + 4)\]\[D = 5(-5) + 0 + 10\]\[D = -25 + 10 = -15\]

Step 3: Calculate Determinant for x (Dx)

Replace the first column of matrix \(A\) with vector \(B\) to form \(A_x\):\[A_x = \begin{bmatrix} 15 & 2 & 1 \ 9 & -1 & 1 \ 13 & 3 & 2 \end{bmatrix}\]Calculate the determinant:\[D_x = \begin{vmatrix} 15 & 2 & 1 \ 9 & -1 & 1 \ 13 & 3 & 2 \end{vmatrix}\]Using the same method:\[D_x = 15((-1)(2) - (1)(3)) - 2((9)(2) - (1)(13)) + 1((9)(3) - (-1)(13))\]\[D_x = 15(-5) - 2(5) + 1(40)\]\[D_x = -75 - 10 + 40 = -45\]

Step 4: Calculate Determinant for y (Dy)

Replace the second column of matrix \(A\) with vector \(B\) to form \(A_y\):\[A_y = \begin{bmatrix} 5 & 15 & 1 \ 2 & 9 & 1 \ 4 & 13 & 2 \end{bmatrix}\]Calculate the determinant:\[D_y = \begin{vmatrix} 5 & 15 & 1 \ 2 & 9 & 1 \ 4 & 13 & 2 \end{vmatrix}\]Using the same method:\[D_y = 5((9)(2) - (1)(13)) - 15((2)(2) - (1)(4)) + 1((2)(13) - (9)(4))\]\[D_y = 5(18 - 13) - 15(4 - 4) + 1(26 - 36)\]\[D_y = 5(5) + 0 - 10 = 25 - 10 = 15\]

Step 5: Calculate Determinant for z (Dz)

Replace the third column of matrix \(A\) with vector \(B\) to form \(A_z\):\[A_z = \begin{bmatrix} 5 & 2 & 15 \ 2 & -1 & 9 \ 4 & 3 & 13 \end{bmatrix}\]Calculate the determinant:\[D_z = \begin{vmatrix} 5 & 2 & 15 \ 2 & -1 & 9 \ 4 & 3 & 13 \end{vmatrix}\]Using the same method:\[D_z = 5((-1)(13) - (9)(3)) - 2((2)(13) - (9)(4)) + 15((2)(3) - (-1)(4))\]\[D_z = 5(-13 - 27) - 2(26 - 36) + 15(6 + 4)\]\[D_z = 5(-40) + 20 + 150\]\[D_z = -200 + 170 = -30\]

Step 6: Use Cramer's Rule to Find x, y, and z

Apply Cramer's Rule:\[x = \frac{D_x}{D} = \frac{-45}{-15} = 3\]\[y = \frac{D_y}{D} = \frac{15}{-15} = -1\]\[z = \frac{D_z}{D} = \frac{-30}{-15} = 2\]

Step 7: Verify the Solution

Verify the solution by substituting \(x = 3\), \(y = -1\), and \(z = 2\) back into the original equations:\[5(3) + 2(-1) + 2 = 15 \quad \text{(True)}\]\[2(3) - (-1) + 2 = 9 \quad \text{(True)}\]\[4(3) + 3(-1) + 2(2) = 13 \quad \text{(True)}\]All equations hold true with the calculated values, confirming the solution is correct.

Key Concepts

Systems of Equations

A system of equations is a set of equations with multiple variables that are solved together. Each equation represents a relation between these variables.
A solution to a system of equations results in values for the variables that make all the equations true simultaneously. Here's why systems of equations matter:

• When solving real-world problems, you often need to find more than one unknown.
• Simplifies complex problems by breaking them down into manageable parts.

In our exercise, we're dealing with a system of three equations with three unknowns: \(x\), \(y\), and \(z\). Each equation has its own component and alters the results of the others due to interdependence. Therefore, our objective is to find a common solution that satisfies all given equations simultaneously. Cramer's Rule is one method to tackle such systems when expressed in matrix form.

Determinants

A determinant is a special number calculated from a square matrix. It plays a crucial role in matrix algebra, especially when using methods like Cramer's Rule. The determinant of a matrix helps determine several important properties:

• Inversions: Only matrices with non-zero determinants can be inverted.
• Linear Dependency: Zero determinants signify dependent sets of linear equations.

In the solution steps, we calculate three determinants:

• \(D\): The determinant of the original coefficient matrix \(A\). It helps us know if the system can be solved using this method.
• \(D_x\), \(D_y\), \(D_z\): Determinants that help find values for \(x\), \(y\), and \(z\) respectively.

Calculating these correctly is essential – it tells us about the solvability of the system and the unique solutions we need to find.

Matrix Algebra

Matrix algebra is a branch of mathematics where matrices are used as mathematical objects with operations similar to regular algebra. In the context of solving systems of equations, working with matrices simplifies calculations. Here’s how:

• Matrix Representation: Systems of linear equations can be neatly expressed as matrix equations like \(AX = B\).
• Operations: Multiplication, transpositions, and inversions are more streamlined with matrices.

In our solution, we use matrix algebra to express the system and then apply determinants to solve for unknowns:
Matrices allow us to streamline calculations and solve complex systems efficiently, especially when dealing with several variables and equations.
For upper-level math, matrices form a foundation for various applications and advanced topics in linear algebra and beyond.

Precalculus

Precalculus serves as a bridge between basic algebra and calculus. It introduces more complex concepts that are foundational for higher mathematics, such as understanding functions, complex numbers, and, importantly, matrices and determinants. Cramer's Rule, seen commonly in precalculus, is a powerful tool when dealing with linear equations. It provides a simple method to find solutions using the properties of determinants. Here's why Cramer's Rule is important in precalculus:

• Tackles systems of equations efficiently without depending on elimination or substitution methods.
• Offers insights into how matrix algebra connects with calculus concepts, such as transformations and mapping.

Leveraging the computational nature of determinants, precalculus prepares students for the extensive problem-solving and analytical thinking needed in calculus and further studies. Emphasizing a structured approach to solving equations reflects real-world applications, where modeling and solving equations form critical parts of scientific and engineering tasks.