Ch. 5 - Systems and Matrices

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

Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 25

Chapter 6, Problem 25

Solve each system by elimination. In systems with fractions, first clear denominators.
6x + 7y + 2 = 0
7x - 6y - 26 = 0

Verified step by step guidance

Rewrite each equation to isolate the constant term on the right side. For the first equation, subtract 2 from both sides to get $6x + 7y = -2$. For the second equation, add 26 to both sides to get $7x - 6y = 26$.

Set up the system of equations for elimination: $6x + 7y = -2$ $7x - 6y = 26$

To eliminate one variable, find a common multiple for the coefficients of either $x$ or $y$. For example, multiply the first equation by 6 and the second equation by 7 to align the coefficients of $y$: $6(6x + 7y) = 6(-2) \Rightarrow 36x + 42y = -12$ $7(7x - 6y) = 7(26) \Rightarrow 49x - 42y = 182$

Add the two new equations to eliminate $y$: $(36x + 42y) + (49x - 42y) = -12 + 182$ This simplifies to $85x = 170$.

Solve for $x$ by dividing both sides by 85, then substitute the value of $x$ back into one of the original equations to solve for $y$.

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Key Concepts

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

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and set up these systems is essential for solving them.

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. This technique often requires multiplying equations by constants to align coefficients before elimination.

Clearing Fractions

When equations contain fractions, multiplying both sides by the least common denominator removes the fractions, simplifying calculations. Clearing denominators helps avoid errors and makes the elimination process more straightforward.

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (3x - 2)/((x + 4)(3x² + 1))

Textbook Question

Find the inverse, if it exists, for each matrix. $[\begin{matrix} 2 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{matrix}]$

Textbook Question

Solve each system, using the method indicated.

5x + 2y = -10

3x - 5y = -6 (Gauss-Jordan)

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Textbook Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

6x - 3y - 4 = 0

3x + 6y - 7= 0

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

$x^{2} + y^{2} = 102$

$x^{2} - y^{2} = 17$

Textbook Question

Evaluate each determinant.

$∣ \begin{matrix} 10 & 2 & 1 \\ - 1 & 4 & 3 \\ - 3 & 8 & 10 \end{matrix} ∣$