Question

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\{\begin{array}{r} {x+2 y+3 z=5} \\ {y-5 z=0} \end{array}\right. $$

Name: Problem 17, Chapter 9: Algebra and Trigonometry
Uploaded: 2020-06-12T06:09:15-08:00
Duration: 2 min 45 s
Channel: Ankit Gupta
Description: Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. { x+2 y+3 z=5 y-5 z=0 .

   Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists.
$$
\left\{\begin{array}{r}
{x+2 y+3 z=5} \\
{y-5 z=0}
\end{array}\right.
$$

Algebra and Trigonometry

Robert Blitzer 6th Edition

Chapter 9, Problem 17

Instant Answer

Solved by Expert Ankit Gupta

06/12/2020

Step 1

The coefficients of $x$, $y$, and $z$ in the first equation are 1, 2, and 3 respectively, and the constant term is 5. In the second equation, $x$ is absent, so its coefficient is 0, the coefficients of $y$ and $z$ are 1 and -5 respectively, and the constant term Show more…

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Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\{\begin{array}{r} {x+2 y+3 z=5} \\ {y-5 z=0} \end{array}\right. $$

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Key Concepts

Free Variables and Parametric Representation

When a system has fewer independent equations than variables, some variables are not uniquely determined. These are called free variables, and the general solution can be expressed in terms of these parameters, capturing all possible solutions to the system.

Systems of Linear Equations

A collection of equations where each equation is linear in the variables. The goal in solving such systems is to find values for the variables that satisfy all the equations simultaneously.

Gaussian Elimination

A method for solving systems of linear equations that involves performing elementary row operations such as row swapping, multiplication by nonzero scalars, and row addition to simplify the system into a triangular or row echelon form, making it easier to solve for the variables.

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