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Chapter 9: Problem 82

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} &4 x-y=0\\\ &2 x+3 y=14 \end{aligned}$$

Short Answer

Expert verified
The solution is \(x = 1\) and \(y = 4\).

Step by step solution

01

Write the system in matrix form

Convert the system of equations into the matrix form. The system of equations is: \(4x - y = 0\) \(2x + 3y = 14\)The coefficient matrix \(A\), variable matrix \(X\), and constant matrix \(B\) are: \[ A = \begin{pmatrix} 4 & -1 \ 2 & 3 \end{pmatrix} \], \[ X = \begin{pmatrix} x \ y \end{pmatrix} \], \[ B = \begin{pmatrix} 0 \ 14 \end{pmatrix} \]
02

Determine the determinant of the coefficient matrix \(D\)

Calculate the determinant of matrix \(A\): \[ D = \begin{vmatrix} 4 & -1 \ 2 & 3 \end{vmatrix} = (4)(3) - (-1)(2) = 12 + 2 = 14 \]Since \(D eq 0\), we can use Cramer's rule.
03

Determine determinant \(D_x\)

Replace the first column of \(A\) with matrix \(B\) and calculate the determinant: \[ A_x = \begin{pmatrix} 0 & -1 \ 14 & 3 \end{pmatrix} \] \[ D_x = \begin{vmatrix} 0 & -1 \ 14 & 3 \end{vmatrix} = (0)(3) - (-1)(14) = 0 + 14 = 14 \]
04

Determine determinant \(D_y\)

Replace the second column of \(A\) with matrix \(B\) and calculate the determinant: \[ A_y = \begin{pmatrix} 4 & 0 \ 2 & 14 \end{pmatrix} \] \[ D_y = \begin{vmatrix} 4 & 0 \ 2 & 14 \end{vmatrix} = (4)(14) - (0)(2) = 56 \]
05

Solve for \(x\) and \(y\)

Using Cramer's rule, solve for \(x\) and \(y\): \[ x = \frac{D_x}{D} = \frac{14}{14} = 1 \] \[ y = \frac{D_y}{D} = \frac{56}{14} = 4 \]Thus, the solution to the system of equations is \(x = 1\) and \(y = 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations involves multiple equations working with the same set of variables. In this problem, we have a system of two linear equations: the first is \(4x - y = 0\), and the second is \(2x + 3y = 14\). These equations need to be solved simultaneously, meaning they will intersect at a common point in the coordinate plane.
It's essential to rewrite the system in a standard form like \(Ax + By = C\). This form makes it easier to transition to matrix notation, a fundamental step in applying methods like Cramer's rule.
Linear systems can be studied using various methods such as substitution, elimination, or advanced algebraic methods like matrix operations.
Matrix Determinant
The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant \(D\) is given by \[D = ad - bc.\]
Determinants are crucial in linear algebra because they give us important information about the matrix. Specifically, the determinant can tell us whether the matrix has an inverse. If \(\|A\|\) 鈮� 0, the matrix is invertible, and we can use Cramer's rule.
In our exercise, the coefficient matrix is: \[A = \begin{pmatrix} 4 & -1 \ 2 & 3 \end{pmatrix}.\]
By calculating the determinant, we found \(D = 14\), indicating that the system has a unique solution.
Solving Linear Equations
Solving linear equations using Cramer's rule involves calculating determinants of matrices derived from the original coefficient matrix. For a system represented by \(AX = B\), where \(A\) is a 2x2 coefficient matrix, \(X\) is the variable matrix, and \(B\) is the constant matrix, the steps are:
  • Compute the determinant \(D\) of the coefficient matrix \(A\).
  • Find the determinant \(Dx\) by replacing the first column of \(A\) with the constant matrix \(B\).
  • Find the determinant \(Dy\) by replacing the second column of \(A\) with the constant matrix \(B\).
  • Use the formulas \(x = \frac{Dx}{D}\) and \(y = \frac{Dy}{D}\) to find the values of \(x\) and \(y\).
In this case, after computing the relevant determinants, we find:
\[ x = \frac{14}{14} = 1, \ y = \frac{56}{14} = 4. \]
Therefore, the solution to the given system of equations is \(x = 1\) and \(y = 4\). This method ensures that we can solve the system accurately as long as the determinant \(D\) is non-zero.

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Most popular questions from this chapter

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{array}{r} x+y=4 \\ 2 x-y=2 \end{array}$$

For each pair of matrices \(A\) and \(B,\) find \((a) A B\) and \((b) B A\). $$A=\left[\begin{array}{rrr} -1 & 0 & 1 \\ 0 & 1 & 1 \\ -1 & -1 & 0 \end{array}\right], B=\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$$

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} &\frac{1}{2} x+\frac{1}{3} y=\frac{49}{18}\\\ &\frac{1}{2} x+2 y=\frac{4}{3} \end{aligned}$$

Solve each problem. Tire Sales The number of automobile tire sales is dependent on several variables. In one study the relationship among annual tire sales \(S\) (in thousands of dollars), automobile registrations \(R\) (in millions), and personal disposable income \(I\) (in millions of dollars) was investigated. The results for three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b R+c l $$ where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Jarrett, J., Business Forecasting Methods, Basil Blackwell, Ltd.) (a) Substitute the values for \(S, R,\) and \(I\) for each year from the table into the equation \(S=a+b R+c I,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. (d) If \(R=117.6\) and \(I=310.73,\) predict \(S .\) (The actual value for \(S\) was \(11,314 .\) ) (e) If \(R=143.8 \text { and } I=829.06, \text { predict } S . \text { (The actual value for } S \text { was } 18,481 .)\) $$\begin{array}{|c|c|c|} \hline S & R & I \\ \hline 10,170 & 112.9 & 307.5 \\\ \hline 15,305 & 132.9 & 621.63 \\ \hline 21,289 & 155.2 & 1937.13 \\\ \hline \end{array}$$

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} 3 x+4 y &=-3 \\ -5 x+8 y &=16 \end{aligned}$$
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