Problem 21 - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations. \(3 a+c=23\) \(4 a+7 b-2 c=-22\) \(8 a-b-c=34\)

Step-by-Step Solution

Answer

The solution is approximately \(a \approx 4.86\), \(b \approx -3.52\), \(c \approx 8.41\).

Step 1: Set Up the Coefficient Matrix

For the system of equations: \[3a + c = 23 \]\[4a + 7b - 2c = -22 \]\[8a - b - c = 34 \]create the coefficient matrix \(A\) associated with the variables \(a\), \(b\), and \(c\):\[A = \begin{bmatrix} 3 & 0 & 1 \ 4 & 7 & -2 \ 8 & -1 & -1 \end{bmatrix}\]

Step 2: Find the Determinant of the Coefficient Matrix

Calculate the determinant of the coefficient matrix \(A\):\[|A| = \begin{vmatrix} 3 & 0 & 1 \ 4 & 7 & -2 \ 8 & -1 & -1 \end{vmatrix}\]Expand the determinant using the first row:\[|A| = 3 \begin{vmatrix} 7 & -2 \ -1 & -1 \end{vmatrix} - 0 + 1 \begin{vmatrix} 4 & 7 \ 8 & -1 \end{vmatrix}\]Evaluate the minors:\[= 3((7)(-1) - (-2)(-1)) + 1((4)(-1) - (7)(8))\]\[= 3(-7 - 2) + 1(-4 - 56)\]\[= 3(-9) + 1(-60) = -27 - 60 = -87 \]

Step 3: Create the Matrix for a (Replace the First Column)

Replace the first column of \(A\) with the constants from the right-hand sides of the equations:\[A_a = \begin{bmatrix} 23 & 0 & 1 \ -22 & 7 & -2 \ 34 & -1 & -1 \end{bmatrix}\]

Step 4: Calculate the Determinant for a

Calculate the determinant of the matrix \(A_a\):\[|A_a| = \begin{vmatrix} 23 & 0 & 1 \ -22 & 7 & -2 \ 34 & -1 & -1 \end{vmatrix}\]Using the first row, find the determinant:\[= 23\begin{vmatrix} 7 & -2 \ -1 & -1 \end{vmatrix} - 0 + 1\begin{vmatrix} -22 & 7 \ 34 & -1 \end{vmatrix}\]Evaluate the minors:\[= 23((7)(-1) - (-2)(-1)) + 1((-22)(-1) - (34)(7))\]\[= 23(-7 - 2) + 1(22 - 238)\]\[= 23(-9) + 1(-216)\]\[= -207 - 216 = -423 \]

Step 5: Solve for Variable a

Using Cramer's Rule, solve for \(a\):\[a = \frac{|A_a|}{|A|} = \frac{-423}{-87} = 4.86 \]

Step 6: Create the Matrix for b (Replace the Second Column)

To find \(b\), replace the second column of \(A\) with the constants:\[A_b = \begin{bmatrix} 3 & 23 & 1 \ 4 & -22 & -2 \ 8 & 34 & -1 \end{bmatrix}\]

Step 7: Calculate the Determinant for b

Calculate the determinant of the matrix \(A_b\):\[|A_b| = \begin{vmatrix} 3 & 23 & 1 \ 4 & -22 & -2 \ 8 & 34 & -1 \end{vmatrix}\]Using the first row, find the determinant:\[= 3\begin{vmatrix} -22 & -2 \ 34 & -1 \end{vmatrix} - 23\begin{vmatrix} 4 & -2 \ 8 & -1 \end{vmatrix} + 1\begin{vmatrix} 4 & -22 \ 8 & 34 \end{vmatrix}\]Evaluate the minors:\[= 3((-22)(-1) - (-2)(34)) - 23((4)(-1) - (-2)(8)) + 1((4)(34) - (-22)(8))\]\[= 3(22 + 68) - 23(-4 + 16) + 1(136 + 176)\]\[= 3(90) - 23(12) + 312\]\[= 270 - 276 + 312 = 306 \]

Step 8: Solve for Variable b

Using Cramer's Rule, solve for \(b\):\[b = \frac{|A_b|}{|A|} = \frac{306}{-87} = -3.52 \]

Step 9: Create the Matrix for c (Replace the Third Column)

Replace the third column of \(A\) with the constants:\[A_c = \begin{bmatrix} 3 & 0 & 23 \ 4 & 7 & -22 \ 8 & -1 & 34 \end{bmatrix}\]

Step 10: Calculate the Determinant for c

Calculate the determinant of the matrix \(A_c\):\[|A_c| = \begin{vmatrix} 3 & 0 & 23 \ 4 & 7 & -22 \ 8 & -1 & 34 \end{vmatrix}\]Using the first row, find the determinant:\[= 3\begin{vmatrix} 7 & -22 \ -1 & 34 \end{vmatrix} - 0 + 23\begin{vmatrix} 4 & 7 \ 8 & -1 \end{vmatrix}\]Evaluate the minors:\[= 3((7)(34) - (-22)(-1)) + 23((4)(-1) - (7)(8))\]\[= 3(238 + 22) + 23(-4 - 56)\]\[= 3(216) - 23(60)\]\[= 648 - 1380 = -732 \]

Step 11: Solve for Variable c

Using Cramer's Rule, solve for \(c\):\[c = \frac{|A_c|}{|A|} = \frac{-732}{-87} = 8.41 \]

Key Concepts

Systems of Equations

When working with systems of equations, you're dealing with multiple equations that share the same set of unknown variables. The goal is to find a common solution for all the variables, making each equation valid simultaneously. In our example, the system consists of three linear equations with three variables: \( a \), \( b \), and \( c \). You might even turn these equations into a more visual form simply by listing them in a set:

• \(3a + c = 23\)
• \(4a + 7b - 2c = -22\)
• \(8a - b - c = 34\)

Solving such systems can be done using various methods, including substitution, elimination, or matrix operations such as Cramer's Rule. Each method comes with its pros and cons, often depending on how complex the equations are. Cramer's Rule, for instance, is particularly useful when you have the same number of equations as unknowns.

Determinants

The determinant is a special number that can be calculated from a square matrix. It provides insights into certain matrix properties, like whether it has an inverse or the volume scaling factor in linear transformations. In Cramer's Rule, determinants play a crucial role in finding specific solutions for equations. Calculating a determinant for a 3x3 matrix involves expanding along one row and finding minors and cofactors—a multi-step process that can get a bit complex. In this exercise, the determinant of the coefficient matrix \(A\):\[ \begin{vmatrix} 3 & 0 & 1 \ 4 & 7 & -2 \ 8 & -1 & -1 \end{vmatrix} = -87 \]This determinant is essential; if it were zero, Cramer's Rule couldn't be used, as the system wouldn't have a unique solution. Fortunately, in our case, the non-zero determinant confirms a unique solution exists for the variables \(a\), \(b\), and \(c\). This is why determinants are so important when solving systems using matrices.

Matrix Operations

In mathematics, matrices are extremely useful for organizing data and performing linear transformations. Matrix operations allow you to transform and solve complex systems more easily. In the context of systems of equations, matrices help streamline processes like solving multiple equations simultaneously. There are several key operations involved when using matrices:

• Setting up the matrix: Extract coefficients from equations to form the coefficient matrix.
• Determinant calculation: As seen with the coefficient matrix \(A\).
• Matrix substitution: Replace columns of the matrix with solution column for each variable.
• Division: After calculating determinants for substituted matrices, division gives the solution for the specific variable.

For our exercise using Cramer's Rule, you start with the coefficient matrix \(A\) and calculate determinants for modified matrices \(A_a\), \(A_b\), and \(A_c\) by systematically replacing columns of \(A\) with the results of the linear equations. Through these operations, you derive the solutions: \(a = 4.86\), \(b = -3.52\), and \(c = 8.41\). Thus, matrix operations make the solving of these complex systems not just possible, but straightforward.