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}⎩⎨⎧2x−4y+3z=17x+2y−z=04x−y−z=6

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 11

6장, 문제 11

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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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 \(\end{cases}\)⎩⎨⎧2x−4y+3z=17x+2y−z=04x−y−z=6