Textbook Question
Solve each problem. Find all values of b such that the straight line 3x - y = b touches the circle x2 + y2 = 25 at only one point.
Ch. 5 - Systems and Matrices
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 6, Problem 63
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.
x + y = 4
2x - y = 2
Verified step by step guidance1
Write the system of equations in standard form: \\ \(x + y = 4\) and \(2x - y = 2\).
Identify the coefficient matrix and calculate the determinant \(D\(: \\ \)D = \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} = (1)(-1) - (1)(2)\).
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} 4 & 1 \\ 2 & -1 \end{vmatrix} = (4)(-1) - (1)(2)\).
Calculate the determinant \(D_y\( by replacing the second column of the coefficient matrix with the constants: \\ \)D_y = \begin{vmatrix} 1 & 4 \\ 2 & 2 \end{vmatrix} = (1)(2) - (4)(2)\).
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. It applies when the system has the same number of equations as unknowns and the determinant of the coefficient matrix (D) is nonzero. The solution for each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.
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Cramer's Rule - 2 Equations with 2 Unknowns
Determinant of a Matrix
The determinant is a scalar value computed from a square matrix that indicates whether the matrix is invertible. For a 2x2 matrix, it is calculated as ad - bc. If the determinant is zero, the system may have infinitely many solutions or no solution, meaning Cramer's Rule cannot be applied directly.
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Alternative Methods for Solving Systems
When the determinant D equals zero, Cramer's Rule fails, so other methods like substitution, elimination, or matrix row reduction are used. These methods help determine if the system has no solution (inconsistent) or infinitely many solutions (dependent). Understanding these alternatives ensures a complete approach to solving linear systems.
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Related Practice
Textbook Question
Solve each problem using a system of equations in two variables. See Example 6. The longest side of a right triangle is 13 m in length. One of the other sides is 7 m longer than the shortest side. Find the lengths of the two shorter sides of the triangle.
Textbook Question
Graph the solution set of each system of inequalities.
Textbook Question
Answer each question. Does the straight line 3x - 2y = 9 intersect the circle x2 + y2 = 25? (Hint: To find out, solve the system formed by these two equations.)
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Textbook Question
Solve each problem. Find the equation of the line passing through the points of intersection of the graphs of x2 + y2 = 20 and x2 - y = 0.
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Textbook Question
Find each product, if possible. See Examples 5–7. <4x2 Matrix>
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