Chapter 10: Problem 30
Solve by determinants. $$\begin{array}{l} y=2 x-3 \\ x=19-3 y \end{array}$$
Short Answer
Expert verified
The solution to the system of equations is \(x = 4\) and \(y = 5\).
Step by step solution
01
Rewrite the equations in standard form
Set each equation so that all the variables and constants are on the left side of the equal sign: Equation 1: y - 2x = -3 Equation 2: 3y + x = 19
02
Write the system as a matrix equation
Create a matrix for the coefficients of the variables and a matrix for the constants: Coefficient matrix: \[\begin{bmatrix} -2 & 1 \ 1 & 3 \end{bmatrix}\] Constant matrix: \[\begin{bmatrix} -3 \ 19 \end{bmatrix}\]
03
Find the determinant of the coefficient matrix
Calculate the determinant of the coefficient matrix (D): \[D = \begin{vmatrix} -2 & 1 \ 1 & 3 \end{vmatrix} = (-2)\cdot(3) - 1\cdot(1) = -6 - 1 = -7\]
04
Find the determinants for x and y
Replace the respective columns in the coefficient matrix with the constant matrix to find the determinants for x (Dx) and y (Dy): \[D_x = \begin{vmatrix} -3 & 1 \ 19 & 3 \end{vmatrix} = (-3)\cdot(3) - 1\cdot(19) = -9 - 19 = -28\] \[D_y = \begin{vmatrix} -2 & -3 \ 1 & 19 \end{vmatrix} = (-2)\cdot(19) - (-3)\cdot(1) = -38 + 3 = -35\]
05
Solve for x and y
Using Cramer's Rule, calculate x and y by dividing Dx and Dy by D: \[x = \frac{D_x}{D} = \frac{-28}{-7} = 4\] \[y = \frac{D_y}{D} = \frac{-35}{-7} = 5\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cramer's Rule
Cramer's Rule is a straightforward method for solving systems of linear equations using determinants. Let's look at how Cramer's Rule applies to a system of equations, as it ultimately lends itself to solving for variables with ease.
When a system of linear equations is given in the standard form, Cramer's Rule can be a powerful tool. With our example system, given by the equations:
- Equation 1: y - 2x = -3
- Equation 2: 3y + x = 19,
we use the coefficients of x and y to form a coefficient matrix and the constants to form a constant matrix. Cramer's Rule then tells us that, provided the determinant of the coefficient matrix is not zero, each variable can be solved by replacing one column of this matrix with the constant matrix and finding the new determinant. The value of the variable is then obtained by dividing this determinant by the determinant of the coefficient matrix.
Using this process we're able to calculate the values of x and y without having to manipulate the original equations extensively. This method is particularly useful for larger systems where traditional elimination or substitution methods might be cumbersome.
When a system of linear equations is given in the standard form, Cramer's Rule can be a powerful tool. With our example system, given by the equations:
- Equation 1: y - 2x = -3
- Equation 2: 3y + x = 19,
we use the coefficients of x and y to form a coefficient matrix and the constants to form a constant matrix. Cramer's Rule then tells us that, provided the determinant of the coefficient matrix is not zero, each variable can be solved by replacing one column of this matrix with the constant matrix and finding the new determinant. The value of the variable is then obtained by dividing this determinant by the determinant of the coefficient matrix.
Using this process we're able to calculate the values of x and y without having to manipulate the original equations extensively. This method is particularly useful for larger systems where traditional elimination or substitution methods might be cumbersome.
Determinants
The determinant of a square matrix is a special scalar value that provides critical information about the matrix and the system it represents. The key properties of determinants make them invaluable in solving systems of equations via methods like Cramer's Rule.
A determinant can tell us if a system has a unique solution (if it's non-zero), an infinite number of solutions (if it's zero and the system is consistent), or no solution (if it's zero and the system is inconsistent). Our focus regarding determinants with Cramer's Rule is based on the fact that they can provide the solution to each variable in a system of equations by adjusting and computing the determinant for each variable's matrix.
In this exercise, we calculate the determinant of the coefficient matrix and make sure it's not zero to proceed. Then, we find the determinants of matrices formed by replacing rows with the constant terms, which correlates directly to solving for the corresponding variable in the equation. It's essential to understand how to compute a determinant because it's the foundation of Cramer's Rule and also plays a role in other areas of linear algebra, like finding the inverse of a matrix.
A determinant can tell us if a system has a unique solution (if it's non-zero), an infinite number of solutions (if it's zero and the system is consistent), or no solution (if it's zero and the system is inconsistent). Our focus regarding determinants with Cramer's Rule is based on the fact that they can provide the solution to each variable in a system of equations by adjusting and computing the determinant for each variable's matrix.
In this exercise, we calculate the determinant of the coefficient matrix and make sure it's not zero to proceed. Then, we find the determinants of matrices formed by replacing rows with the constant terms, which correlates directly to solving for the corresponding variable in the equation. It's essential to understand how to compute a determinant because it's the foundation of Cramer's Rule and also plays a role in other areas of linear algebra, like finding the inverse of a matrix.
Matrix Equation
A matrix equation is a representation of a system of linear equations using matrices. It transforms the equations into a compact form, simplifying operations and providing a framework for applying linear algebra techniques.
In our example, we converted the equations into a matrix equation by isolating the coefficients of the variables and the constant terms. This process resulted in a coefficient matrix and a constant matrix. We then used these matrices to solve the system using Cramer's Rule, showcasing the utility of a matrix equation. A matrix equation also allows the use of matrix operations such as multiplication and finding inverses, which are essential in other methods of solving systems of equations, such as the matrix inverse method.
It's important to be familiar with writing systems as matrix equations because this step is fundamental in applying various analytical techniques, not just in algebra, but also across other disciplines like economics, engineering, and sciences that utilize linear models.
In our example, we converted the equations into a matrix equation by isolating the coefficients of the variables and the constant terms. This process resulted in a coefficient matrix and a constant matrix. We then used these matrices to solve the system using Cramer's Rule, showcasing the utility of a matrix equation. A matrix equation also allows the use of matrix operations such as multiplication and finding inverses, which are essential in other methods of solving systems of equations, such as the matrix inverse method.
It's important to be familiar with writing systems as matrix equations because this step is fundamental in applying various analytical techniques, not just in algebra, but also across other disciplines like economics, engineering, and sciences that utilize linear models.
