Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 67

Chapter 6, Problem 67

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.
$5 x + 4 y = 10$
$5 x + 4 y = 10$

Verified step by step guidance

Write the system of equations in standard form: $5x + 4y = 10$ and $3x - 7y = 6$.

Identify the coefficient matrix and calculate the determinant $D$: $D = \begin{vmatrix} 5 & 4 \\ 3 & -7 \end{vmatrix} = (5)(-7) - (4)(3)$.

Calculate the determinant $D_x$ by replacing the first column of the coefficient matrix with the constants from the right side of the equations: $D_x = \begin{vmatrix} 10 & 4 \\ 6 & -7 \end{vmatrix} = (10)(-7) - (4)(6)$.

Calculate the determinant $D_y$ by replacing the second column of the coefficient matrix with the constants: $D_y = \begin{vmatrix} 5 & 10 \\ 3 & 6 \end{vmatrix} = (5)(6) - (10)(3)$.

Use Cramer's rule to find the solutions for $x$ and $y$: $x = \frac{D_x}{D}$ and $y = \frac{D_y}{D}$. If $D = 0$, then 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 with two variables, 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.

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 of equations has a unique solution (nonzero determinant) or if it is dependent or inconsistent (zero determinant). It is essential for applying Cramer's Rule.

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 matrix row reduction are used to analyze the system and find the solution set or determine inconsistency.

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Related Practice

Textbook Question

Graph the solution set of each system of inequalities.

$e^{x}-y$\le$1$

$x-2y$\ge$4$

576

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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.

4x + 3y = -7

2x + 3y = -11

618

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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. 1.5

x + 3y = 5

2x + 4y = 3

613

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Textbook Question

Solve each problem. Find the radius and height (to the nearest thousandth) of an open-ended cylinder with volume 50 in.³ and lateral surface area 65 in.².

451

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Textbook Question

Graph the solution set of each system of inequalities.

$y \leq \log x$

$y \geq ∣ x - 2 ∣$

531

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Textbook Question

Find each product, if possible.

$[\begin{matrix} - 2 & - 3 & - 4 \\ 2 & - 1 & 0 \\ 4 & - 2 & 3 \end{matrix}] [\begin{matrix} 0 & 1 & 4 \\ 1 & 2 & - 1 \\ 3 & 2 & - 2 \end{matrix}]$