Problem 44
Question
Write the system of linear equations for which Cramer's Rule yields the given determinants. $$D=\left|\begin{array}{rr}{2} & {-3} \\\\{5} & {6}\end{array}\right|, \quad D_{x}=\left|\begin{array}{rr} {8} & {-3} \\\\{11} & {6}\end{array}\right|$$
Step-by-Step Solution
Verified Answer
The system of linear equations, taking into account the values from the determinants, is: \[2x - 3y = 8\]\[5x + 6y = 11\]
1Step 1: Analyze the given determinants
Two determinants are given: \(D\), which is the determinant of the coefficient matrix, and \(D_{x}\), where the first column of the coefficient matrix is replaced with the constants of the equations. This information will be used to derive the system of linear equations. The determinant of a 2x2 matrix \(\left|\begin{array}{cc}a & b \\ c & d\end{array}\right|\) is calculated as \(ad - bc\).
2Step 2: Identify the coefficients and constants
By comparing the given determinants with the typical structure of a determinant for a system of linear equations, identify the coefficients of the variables and the constants in the equations. From \(D\), the coefficients of the variables in the system of equations are \(2, -3\) for the first equation and \(5, 6\) for the second equation. From \(D_{x}\), the constants in the equations are \(8, 11\). Hence, the system of linear equations is given by: \[2x - 3y = 8\]\[5x + 6y = 11\]
Key Concepts
System of Linear EquationsDeterminantsCoefficient Matrix
System of Linear Equations
Understanding a system of linear equations is the starting point for solving any linear problem algebraically. At its core, such a system consists of two or more linear equations with the same set of variables. Think of it as a quest to find the precise values for variables that make all the equations true simultaneously.
These sets of equations represent lines, planes, or hyperplanes in geometry, which coincide at a point or a line – the solution to the system. When we apply methods such as substitution, elimination, or tools like Cramer's Rule, we're essentially strategizing to unearth where these linear representations meet, if they indeed do.
In a practical scenario, say you're trying to find out how many pencils and erasers you can buy with a limited amount of money. Each item's cost creates a linear relationship: the more items, the higher the cost. If you combine this with another restriction, like the space in your bag, you're dealing with a system of linear equations where the solutions tell you exactly how many of each item you can take.
These sets of equations represent lines, planes, or hyperplanes in geometry, which coincide at a point or a line – the solution to the system. When we apply methods such as substitution, elimination, or tools like Cramer's Rule, we're essentially strategizing to unearth where these linear representations meet, if they indeed do.
In a practical scenario, say you're trying to find out how many pencils and erasers you can buy with a limited amount of money. Each item's cost creates a linear relationship: the more items, the higher the cost. If you combine this with another restriction, like the space in your bag, you're dealing with a system of linear equations where the solutions tell you exactly how many of each item you can take.
Determinants
The concept of determinants plays a pivotal role in linear algebra, particularly when working with square matrices. A determinant is a special number that can be calculated from a square matrix. Intuitively, the determinant gives us information about the properties of the matrix, including whether the system has a unique solution, no solution, or infinitely many solutions.
For a simple 2x2 matrix, the determinant is found by subtracting the product of the off-diagonal elements from the product of the diagonal elements, expressed as \[D = ad - bc\] if the matrix looks like \[\begin{array}{cc}a & b \ c & d\end{array}\] The values of determinants answer critical questions: if the determinant of the coefficient matrix in a linear system is non-zero, it heralds a unique solution; if it's zero, it warns of either no solutions or an infinite number of them.
It's fascinating that such a simple computation opens the gates to knowing whether a system's (mathematical or real-life) outcome will be solvable and straightforward or complex and multifaceted. Determinants aren't just abstract entities; they're the contextual bridge between numbers and the graphical stories they tell.
For a simple 2x2 matrix, the determinant is found by subtracting the product of the off-diagonal elements from the product of the diagonal elements, expressed as \[D = ad - bc\] if the matrix looks like \[\begin{array}{cc}a & b \ c & d\end{array}\] The values of determinants answer critical questions: if the determinant of the coefficient matrix in a linear system is non-zero, it heralds a unique solution; if it's zero, it warns of either no solutions or an infinite number of them.
It's fascinating that such a simple computation opens the gates to knowing whether a system's (mathematical or real-life) outcome will be solvable and straightforward or complex and multifaceted. Determinants aren't just abstract entities; they're the contextual bridge between numbers and the graphical stories they tell.
Coefficient Matrix
The coefficient matrix is the cornerstone of understanding systems of linear equations in matrix form. It's a neatly organized framework where we capture all the coefficients of the variables from our system. Essentially, each row corresponds to a linear equation while each column aligns with a particular variable.
When confronted with a system such as \[ax + by = e\]\[cx + dy = f\], we can extract the coefficient matrix as \[\begin{array}{cc} a & b \ c & d \end{array}\]. This matrix serves as a skeletal structure holding the system together, dictating relationships between variables, and awaiting the input of values. When we sling the constants (\(e, f\)) into the mix, the coefficient matrix transforms into an augmented matrix that encompasses the entire system.
In the case of Cramer's Rule, the coefficient matrix is pivotal: it's the backbone of our calculations. By evaluating its determinant, we estimate whether a unique solution exists, resembling a battle strategist contemplating various routes - each determinant you calculate offers insights and guides you to the next best move.
When confronted with a system such as \[ax + by = e\]\[cx + dy = f\], we can extract the coefficient matrix as \[\begin{array}{cc} a & b \ c & d \end{array}\]. This matrix serves as a skeletal structure holding the system together, dictating relationships between variables, and awaiting the input of values. When we sling the constants (\(e, f\)) into the mix, the coefficient matrix transforms into an augmented matrix that encompasses the entire system.
In the case of Cramer's Rule, the coefficient matrix is pivotal: it's the backbone of our calculations. By evaluating its determinant, we estimate whether a unique solution exists, resembling a battle strategist contemplating various routes - each determinant you calculate offers insights and guides you to the next best move.
Other exercises in this chapter
Problem 44
Solve the system: $$ \left\\{\begin{aligned} \ln w+\ln x+\ln y+& \ln z &=-1 \\ -\ln w+4 \ln x+\ln y-& \ln z=& 0 \\ \ln w-2 \ln x+\ln y-& 2 \ln z=& 11 \\ -\ln w-
View solution Problem 44
Determine whether each statement makes sense or does not make sense, and explain your reasoning. Models for controlling traffic flow are based on an equal numbe
View solution Problem 44
Find \(A^{-1}\) and check. $$ A=\left[\begin{array}{cc} {e^{2 x}} & {-e^{x}} \\ {e^{3 x}} & {e^{2 x}} \end{array}\right] $$
View solution Problem 45
Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array
View solution