Problem 20
Question
Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{l}3 x=7 y+1 \\\2 x=3 y-1\end{array}\right.$$
Step-by-Step Solution
Verified Answer
The solution to the system of equations is \(x = -2\) and \(y = -1\)
1Step 1: Formulate coefficient and solutions matrices
The coefficient matrix is \(A = \left[\begin{array}{ll} 3 & -7 \\ 2 & -3\end{array}\right]\), and the solutions matrix is \(B = \left[\begin{array}{c} 1 \\ -1\end{array}\right]\).
2Step 2: Compute the determinant of A
Compute the determinant of A, denoted as \(\|A\|\). This is done by using the main formula for 2 x 2 matrices: \( \|A\| = (\text{ad} - \text{bc}) = (3*-3) - (2*-7) = -9 - -14 = 5 .\)
3Step 3: Compute the determinant of A1 and A2
Compute the determinants of A1 and A2, the matrices that arise from replacing the first and second columns of A, respectively, with the matrix B. For A1, the replacement gives: \(A1 = \left[\begin{array}{ll} 1 & -7 \\ -1 & -3\end{array}\right]\). Thus, the determinant, \(\|A1\|\) is: \(\|A1\| = (1*-3) - ( -1*-7) = -3 - 7 = -10.\ For A2, the replacement gives: \(A2 = \left[\begin{array}{ll} 3 & 1 \\ 2 & -1\end{array}\right]\). Thus, the determinant, \(\|A2\|\) is: \(\|A2\| = (3*-1)-(2*1) = -5.\
4Step 4: Solve for the variables
Finally, use the determinants to solve for the variables x and y using the formula \(x = \|A1\| / \|A\|\) and \(y = \|A2\| / \|A\|\) . So, \(x = -10 / 5 = -2\) , and \(y = -5 / 5 = -1\) .
Key Concepts
Understanding Systems of EquationsDeterminant CalculationMatrix Algebra and Cramer's Rule
Understanding Systems of Equations
Solving systems of equations is a foundational skill in algebra that allows you to find the values of variables that satisfy multiple equations at the same time. In the context of our exercise, we have a system of two linear equations with two variables, which can be written in the general form:
\[ \begin{align*} ax + by &= e\ cx + dy &= f \end{align*} \]
Here, the goal is to solve for the unknowns ‘x’ and ‘y’. One of the most efficient ways to solve such a system is using Cramer's Rule. This method relies on determinant calculation and matrix algebra to find the solution. The major advantage of Cramer's Rule is that it gives an explicit solution to the variables if the system has a unique solution, which means that the determinant of the coefficient matrix is not zero. Also, Cramer's Rule is applicable only when the system has the same number of equations as variables. To improve upon the understanding, try to set up the system in matrix form and practice the determinant calculation separately before applying the rule.
\[ \begin{align*} ax + by &= e\ cx + dy &= f \end{align*} \]
Here, the goal is to solve for the unknowns ‘x’ and ‘y’. One of the most efficient ways to solve such a system is using Cramer's Rule. This method relies on determinant calculation and matrix algebra to find the solution. The major advantage of Cramer's Rule is that it gives an explicit solution to the variables if the system has a unique solution, which means that the determinant of the coefficient matrix is not zero. Also, Cramer's Rule is applicable only when the system has the same number of equations as variables. To improve upon the understanding, try to set up the system in matrix form and practice the determinant calculation separately before applying the rule.
Determinant Calculation
The determinant of a matrix is a special scalar value that can be computed from its elements and provides important information about the matrix, such as whether it's invertible or not. In our exercise, we encountered a 2x2 matrix, which has a straightforward formula for determining its determinant:
\[ \text{If } A = \begin{bmatrix} a & b\ c & d \end{bmatrix}, \text{then } |A| = ad - bc \]
It is extremely helpful to grasp the geometric meaning of a determinant; for a 2x2 matrix, it represents the area of the parallelogram formed by the vectors of the matrix’s columns. For larger matrices, the process becomes more complex, involving minors and cofactors, but the basic concept is similar - the determinant can tell you about the linear independence of the system and if a unique solution is possible. Memorizing determinant formulas and practicing them on different matrices will bolster your problem-solving speed and accuracy.
\[ \text{If } A = \begin{bmatrix} a & b\ c & d \end{bmatrix}, \text{then } |A| = ad - bc \]
It is extremely helpful to grasp the geometric meaning of a determinant; for a 2x2 matrix, it represents the area of the parallelogram formed by the vectors of the matrix’s columns. For larger matrices, the process becomes more complex, involving minors and cofactors, but the basic concept is similar - the determinant can tell you about the linear independence of the system and if a unique solution is possible. Memorizing determinant formulas and practicing them on different matrices will bolster your problem-solving speed and accuracy.
Matrix Algebra and Cramer's Rule
Matrix algebra involves operations such as addition, subtraction, multiplication, and finding the inverse of matrices. For solving systems of equations, Cramer's Rule uses matrix algebra as its groundwork. The rule states that for a system of linear equations to be solved, each variable is equal to the determinant of a matrix divided by the determinant of the coefficient matrix. Specifically, in our example:
\[ x = \frac{|A1|}{|A|}, \quad y = \frac{|A2|}{|A|} \]
Here, matrices \(A1\) and \(A2\) are formed by replacing the respective columns of the coefficient matrix with the solution matrix. To gain full command of Cramer's Rule, you should become comfortable with matrix operations, especially with the concept of replacing columns to form new matrices for determinant calculation. Cramer's Rule shines because of its algorithmic nature, which, while not always the most efficient for large systems, offers clarity and a systematic approach for smaller ones. Simplifying and practicing these steps with various systems can deepen your understanding and enhance your matrix algebra skills.
\[ x = \frac{|A1|}{|A|}, \quad y = \frac{|A2|}{|A|} \]
Here, matrices \(A1\) and \(A2\) are formed by replacing the respective columns of the coefficient matrix with the solution matrix. To gain full command of Cramer's Rule, you should become comfortable with matrix operations, especially with the concept of replacing columns to form new matrices for determinant calculation. Cramer's Rule shines because of its algorithmic nature, which, while not always the most efficient for large systems, offers clarity and a systematic approach for smaller ones. Simplifying and practicing these steps with various systems can deepen your understanding and enhance your matrix algebra skills.
Other exercises in this chapter
Problem 20
In Exercises \(17-26,\) let $$A=\left[\begin{array}{rr}-3 & -7 \\\2 & -9 \\\5 & 0\end{array}\right] \text { and } B=\left[\begin{array}{rr}-5 & -1 \\\0 & 0 \\\3
View solution Problem 20
In Exercises \(19-28,\) find \(A^{-1}\) by forming \([A | I]\) and then using row operations to obtain \([I | B],\) where \(A^{-1}=[B]\). Check that \(A A^{-1}=
View solution Problem 21
In Exercises \(19-28,\) find \(A^{-1}\) by forming \([A | I]\) and then using row operations to obtain \([I | B],\) where \(A^{-1}=[B]\). Check that \(A A^{-1}=
View solution Problem 21
Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{l}2 x=3 y+2 \\\5 x=51-4 y\end{array}\right.$$
View solution