In Exercises 11–14, perform the row operation and write the...

In Exercises 11–14, perform the row operation and write the equivalent system.
Add $-2$ times Equation 1 to Equation 3 .
$$
\left\{\begin{aligned} x-2 y+3 z =5 \\-x+3 y- 5 z=4 \\ 2 x\quad \quad-3 z=0 \end{aligned}\right.
$$
What did this operation accomplish?

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Key Concepts

Row Operations

Row operations are algebraic manipulations performed on the rows of a system of linear equations or their corresponding augmented matrix. They include actions such as swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. These operations are fundamental because they allow the transformation of a system into a simpler equivalent system without altering its solution set.

Elimination Method

The elimination method involves combining equations in order to cancel out one or more variables. By adding, subtracting, or otherwise combining equations, you simplify the system step-by-step, eventually reducing it to a form where the variables can be solved easily. This method is a cornerstone in solving systems of linear equations.

Equivalent Systems

An equivalent system is a reformulated version of the original system of equations that has the same solution set. Through valid row operations, any system can be transformed into an equivalent system that is often easier to analyze and solve. Ensuring equivalence is crucial because it guarantees that the process of simplifying the system does not change the underlying relationships between the variables.

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