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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 73
Chapter 6, Problem 73

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.
(1/2)x + (1/3)y = 2
(3/2)x - (1/2)y = -12

Verified step by step guidance
1
Write the system of equations in standard form: \[\frac{1}{2}x + \frac{1}{3}y = 2\] \[\frac{3}{2}x - \frac{1}{2}y = -12\]
Identify the coefficients for the variables to form the coefficient matrix: \[A = \begin{bmatrix} \frac{1}{2} & \frac{1}{3} \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}\]
Calculate the determinant of matrix \(A\), denoted as \(D\), using the formula: \[D = a_{11}a_{22} - a_{12}a_{21}\] where \(a_{11} = \frac{1}{2}\), \(a_{12} = \frac{1}{3}\), \(a_{21} = \frac{3}{2}\), and \(a_{22} = -\frac{1}{2}\).
If \(D \neq 0\), find determinants \(D_x\) and \(D_y\) by replacing the respective columns of \(A\) with the constants vector: \[D_x = \begin{vmatrix} 2 & \frac{1}{3} \\ -12 & -\frac{1}{2} \end{vmatrix}, \quad D_y = \begin{vmatrix} \frac{1}{2} & 2 \\ \frac{3}{2} & -12 \end{vmatrix}\]
Solve for \(x\) and \(y\) using Cramer's rule: \[x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}\] If \(D = 0\), use another method such as substitution or elimination to find the solution set.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. For a system of two equations, it involves calculating the determinant of the coefficient matrix (D) and determinants of matrices formed by replacing columns with constants. If D ≠ 0, the system has a unique solution found by dividing these determinants by D.
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Cramer's Rule - 2 Equations with 2 Unknowns

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This value helps determine if the system has a unique solution (nonzero determinant) or if it is dependent or inconsistent (zero determinant). It is essential for applying Cramer's Rule correctly.
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Alternative Methods for Solving Systems When D = 0

If the determinant D equals zero, Cramer's Rule cannot be used because the system may have infinitely many solutions or no solution. In such cases, methods like substitution, elimination, or analyzing the system's consistency are used to find the solution set or determine if no solution exists.
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Related Practice
Textbook Question

Given A=[4−231],B=[510−237]A = \(\left\)[ \(\begin{matrix}\) 4 & -2 \\ 3 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 5 & 1 \\ 0 & -2 \\ 3 & 7 \(\end{matrix}\) \(\right\)], and C=[−541036]C = \(\left\)[ \(\begin{matrix}\) -5 & 4 & 1 \\ 0 & 3 & 6 \(\end{matrix}\) \(\right\)], find each product, if possible. See Examples 5–7. AB

Textbook Question

Perform each operation, if possible.

[258192]−[3471]\(\left\)[ \(\begin{matrix}\) 2 & 5 & 8 \\ 1 & 9 & 2 \(\end{matrix}\) \(\right\)] - \(\left\)[ \(\begin{matrix}\) 3 & 4 \\ 7 & 1 \(\end{matrix}\) \(\right\)]

Textbook Question

Solve each system. (Hint: In Exercises 69–72, let 1/x=t1/x = t and 1/y=u1/y = u.)

2x+3y=18\(\frac{2}{x}\)+\(\frac{3}{y}\)=18

4x−5y=−8\(\frac{4}{x}\)-\(\frac{5}{y}\)=-8

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Textbook Question

For what value(s) of k will the following system of linear equations have no solution? infinitely many solutions?

x - 2y = 3

-2x + 4y = k

Textbook Question

Perform each operation, if possible.

[325]−[8−46]+[102]\(\left\)[\(\begin{matrix}\)3\\ 2\\ 5\(\end{matrix}\]\right\)]-\(\left\)[\(\begin{matrix}\)8\\ -4\\ 6\(\end{matrix}\[\right\)]+\(\left\)[\(\begin{matrix}\)1\\ 0\\ 2\(\end{matrix}\]\right\)]

Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

2x - y + 4z = -2

3x + 2y - z = -3

x + 4y - 2z = 17