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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 9
Chapter 6, Problem 9

Solve each system in Exercises 5–18. {3x+2y−3z=−22x−5y+2z=−24x−3y+4z=10\(\begin{cases}\)3x + 2y - 3z = -2 \\2x - 5y + 2z = -2 \\4x - 3y + 4z = 10\(\end{cases}\)

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Write down the system of equations clearly: \[3x + 2y - 3z = -2\] \[2x - 5y + 2z = -2\] \[4x - 3y + 4z = 10\]
Choose a method to solve the system: substitution, elimination, or matrix methods. Here, elimination or substitution are common choices for three variables.
Use elimination or substitution to reduce the system from three equations with three variables to two equations with two variables. For example, eliminate one variable (like \(z\)) by combining pairs of equations.
Solve the resulting two-variable system for two variables (say \(x\) and \(y\)) using substitution or elimination.
Substitute the values of \(x\) and \(y\) back into one of the original equations to solve for \(z\), completing the solution for \((x, y, z)\).

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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 multiple linear equations involving the same set of 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.
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Methods for Solving Systems (Substitution, Elimination, and Matrix Methods)

Common techniques to solve systems include substitution, elimination, and using matrices (such as Gaussian elimination). These methods transform the system into simpler forms to find the values of variables efficiently.
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Consistency and Types of Solutions

Systems can have one unique solution, infinitely many solutions, or no solution. Recognizing the system's consistency helps determine the nature of the solution set and guides the solving process.
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Related Practice
Textbook Question

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 1/x(x-1)

Textbook Question

In Exercises 1–18, solve each system by the substitution method. {xy=62x−y=1\(\begin{cases}\)xy = 6 \\2x - y = 1\(\end{cases}\)

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Textbook Question

In Exercises 5–18, solve each system by the substitution method. {x=4y−2x=6y+8\(\begin{cases}\)x = 4y - 2 \(\x\) = 6y + 8\(\end{cases}\)

Textbook Question

The perimeter of a table tennis top is 28 feet. The difference between 4 times the length and 3 times the width is 21 feet. Find the dimensions.

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Textbook Question

Graph each inequality. x≤−3

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Textbook Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

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