Problem 43
Question
Write the system of linear equations for which Cramer's Rule yields the given determinants. $$ D=\left|\begin{array}{rr} {2} & {-4} \\ {3} & {5} \end{array}\right|, \quad D_{x}=\left|\begin{array}{rr} {8} & {-4} \\ {-10} & {5} \end{array}\right| $$
Step-by-Step Solution
Verified Answer
The system of linear equations for which Cramer's Rule yields the given determinants is 2x - 4y = 8 and 3x + 5y = -10.
1Step 1: Understand the Variables from Coefficients Matrix
The coefficients matrix \(D=\left|\begin{array}{rr} {2} & {-4} \ {3} & {5} \end{array}\right|\) represents the coefficients of two equations in terms of x and y. This can be rewritten in terms of the variables x and y as follows: 2x - 4y = a and 3x + 5y = b.
2Step 2: Solve for the Right Side of the Equation Set
The second determinant given is \(D_x=\left|\begin{array}{rr} {8} & {-4} \ {-10} & {5} \end{array}\right|\). This results when we replace the first column of our initial determinant (D) with the right side of our equations. So, the right side of the equations (a and b) are 8 and -10, respectively.
3Step 3: Create the System of Equations
The final step involves substituting the values 8 and -10 for a and b in the original equations derived in step 1. The system of linear equations formed is 2x - 4y = 8 and 3x + 5y = -10.
Key Concepts
System of Linear EquationsDeterminantsCoefficients MatrixAlgebra
System of Linear Equations
A system of linear equations consists of two or more equations with two or more variables. Each equation represents a straight line when graphed, and the solution to the system is the intersection point of the lines. These equations often describe relationships between quantities and can be expressed in a standard form, such as \( ax + by = c \).
In the original exercise, we explored a system made up of two linear equations: \( 2x - 4y = 8 \) and \( 3x + 5y = -10 \). Solutions to such systems can be found using various methods, including substitution, elimination, or Cramer's Rule, each providing unique perspectives on the relationships between the variables.
Understanding the structure of such systems is crucial for solving problems involving interconnected linear relationships. Each equation brings us closer to understanding how variables are interdependent in their behavior under given circumstances.
In the original exercise, we explored a system made up of two linear equations: \( 2x - 4y = 8 \) and \( 3x + 5y = -10 \). Solutions to such systems can be found using various methods, including substitution, elimination, or Cramer's Rule, each providing unique perspectives on the relationships between the variables.
Understanding the structure of such systems is crucial for solving problems involving interconnected linear relationships. Each equation brings us closer to understanding how variables are interdependent in their behavior under given circumstances.
Determinants
Determinants are values calculated from a special set of numbers organized into a square matrix. In the context of a 2x2 matrix, the determinant is found using the formula \( ad - bc \) for a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
Determinants help in understanding whether a system of equations has a unique solution, no solution, or infinitely many solutions. A non-zero determinant indicates a unique solution exists.
In the exercise, the given determinant \( D \) was calculated as \( 2 \times 5 - (-4) \times 3 = 10 + 12 = 22 \), confirming the existence of potential solutions for the system under Cramer's Rule, where the determinant plays a key role in finding these solutions by altering columns of the coefficients matrix to find specific variable values.
Determinants help in understanding whether a system of equations has a unique solution, no solution, or infinitely many solutions. A non-zero determinant indicates a unique solution exists.
In the exercise, the given determinant \( D \) was calculated as \( 2 \times 5 - (-4) \times 3 = 10 + 12 = 22 \), confirming the existence of potential solutions for the system under Cramer's Rule, where the determinant plays a key role in finding these solutions by altering columns of the coefficients matrix to find specific variable values.
Coefficients Matrix
The coefficients matrix is a compact representation of the coefficients of the variables from a system of linear equations. For two linear equations in the form \( ax + by = c \) and \( dx + ey = f \), the coefficients matrix would be \( \begin{bmatrix} a & b \ d & e \end{bmatrix} \). It essentially organizes the coefficients into a structured form that facilitates various solving techniques.
In the provided exercise, the coefficients matrix was \( \begin{bmatrix} 2 & -4 \ 3 & 5 \end{bmatrix} \). This matrix plays a critical role when utilizing Cramer's Rule, as it allows us to swap columns to focus on solving for particular variables. The creation and manipulation of this matrix are foundational to leveraging algebraic solutions in systems of linear equations.
In the provided exercise, the coefficients matrix was \( \begin{bmatrix} 2 & -4 \ 3 & 5 \end{bmatrix} \). This matrix plays a critical role when utilizing Cramer's Rule, as it allows us to swap columns to focus on solving for particular variables. The creation and manipulation of this matrix are foundational to leveraging algebraic solutions in systems of linear equations.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It allows for the representation and exploration of numeric relationships and changes. In algebra, we use letters to represent numbers, which enables us to formulate equations and solve for unknowns within these equations.
Cramer's Rule, which is rooted in algebra, is a technique specifically designed to solve systems of linear equations by using determinants. We use algebraic manipulation to express variables in terms of others and to derive solutions to equations representing real-world problems.
This exercise illustrates algebraic techniques by manipulating expressions, creating a system of linear equations, and applying these methods to solve practical mathematical problems. By understanding algebra, students gain insights into how mathematical models can represent dynamic systems and solve complex problems.
Cramer's Rule, which is rooted in algebra, is a technique specifically designed to solve systems of linear equations by using determinants. We use algebraic manipulation to express variables in terms of others and to derive solutions to equations representing real-world problems.
This exercise illustrates algebraic techniques by manipulating expressions, creating a system of linear equations, and applying these methods to solve practical mathematical problems. By understanding algebra, students gain insights into how mathematical models can represent dynamic systems and solve complex problems.
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