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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 11
Chapter 6, Problem 11

Solve each system in Exercises 5–18. {2x−4y+3z=17x+2y−z=04x−y−z=6\(\begin{cases}\)2x - 4y + 3z = 17 \(\x\) + 2y - z = 0 \\4x - y - z = 6\(\end{cases}\)

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Write down the system of equations clearly: \[2x - 4y + 3z = 17\] \[x + 2y - z = 0\] \[4x - y - z = 6\]
Choose one equation to express one variable in terms of the others. For example, from the second equation, solve for \[x\]: \[x = -2y + z\]
Substitute the expression for \[x\] into the first and third equations to eliminate \[x\], resulting in two equations with only \[y\] and \[z\].
Simplify these two new equations and solve the resulting system of two equations with two variables (\[y\] and \[z\]) using either substitution or elimination.
Once you find \[y\] and \[z\], substitute these values back into the expression for \[x\] to find the value of \[x\].

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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. Such systems can have one solution, infinitely many solutions, or no solution.
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Methods for Solving Systems (Substitution, Elimination, and Matrix Methods)

Common methods to solve systems include substitution, elimination, and using matrices (such as Gaussian elimination). These techniques transform the system into simpler forms to isolate variables and find their values efficiently.
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Three-Variable Systems

When a system has three variables, it typically involves three equations. Solving requires careful manipulation to reduce the system step-by-step, often by eliminating variables pairwise until a single variable can be solved, then back-substituting to find others.
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In Exercises 5–18, solve each system by the substitution method. 5x + 2y = 0 x - 3y = 0

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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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