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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 11
Capitolo 6, Problema 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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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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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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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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