Solve each system in Exercises 5–18. {x+y+2z=11x+y+3z=14x+2y−z=5\begin{cases} x + y + 2z = 11 \\ x + y + 3z = 14 \\ x + 2y - z = 5 \end{cases}⎩⎨⎧x+y+2z=11x+y+3z=14x+2y−z=5

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 5

6장, 문제 5

Solve each system in Exercises 5–18.
$\begin{cases}\) x + y + 2z = 11 \\ x + y + 3z = 14 \\ x + 2y - z = 5 \(\end{cases}$

검증된 단계별 안내

Write down the system of equations clearly: $\begin{cases} x + 0y + 2z = 11 \\ x + 0y + 3z = 14 \\ x + 2y - 0z = 5 \end{cases}$

Notice that the first two equations both have $x$ and $z$ terms but no $y$. Use these two equations to eliminate $x$ or $z$ by subtracting one equation from the other.

Subtract the first equation from the second: $ (x + 0y + 3z) - (x + 0y + 2z) = 14 - 11 $ which simplifies to an equation involving only $z$.

Solve the resulting equation for $z$. Once you have $z$, substitute this value back into one of the first two equations to solve for $x$.

With $x$ and $z$ known, substitute both into the third equation $x + 2y = 5$ to solve for $y$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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. Understanding how to interpret and set up these systems is essential for solving them.

Methods for Solving Systems

Common methods to solve systems include substitution, elimination, and matrix techniques like Gaussian elimination. Choosing an appropriate method depends on the system's structure. For example, elimination is useful when variables can be easily canceled by adding or subtracting equations.

Interpreting Coefficients and Variables

Coefficients represent the numerical multipliers of variables in equations. Recognizing zero coefficients helps simplify the system by reducing the number of variables in certain equations. This understanding aids in selecting the best approach to isolate variables and solve the system efficiently.

Solve each system in Exercises 5–18. {x+y+2z=11x+y+3z=14x+2y−z=5\(\begin{cases}\) x + y + 2z = 11 \\ x + y + 3z = 14 \\ x + 2y - z = 5 \(\end{cases}\)⎩⎨⎧x+y+2z=11x+y+3z=14x+2y−z=5