Consider the following two systems of equations: 5 x1+x2-3 x3 =0 5 x1+x2-3 x3 =0

step 1 In this problem we're given two sets of equations. So in metrics 4 we can write these equations

Consider the following two systems of equations: 5...

Consider the following two systems of equations: $$ \begin{array}{rlrl}{5 x_{1}+x_{2}-3 x_{3}} & {=0} & {5 x_{1}+x_{2}-3 x_{3}} & {=0} \\ {-9 x_{1}+2 x_{2}+5 x_{3}} & {=1} & {-9 x_{1}+2 x_{2}+5 x_{3}} & {=5} \\ {4 x_{1}+x_{2}-6 x_{3}} & {=9} & {4 x_{1}+x_{2}-6 x_{3}} & {=45}\end{array} $$ It can be shown that the first system has a solution. Use this fact and the theory from this section to explain why the second system must also have a solution. Make no row operations.)

Consider the following two systems of equations:
$$
\begin{array}{rlrl}{5 x_{1}+x_{2}-3 x_{3}} & {=0} & {5 x_{1}+x_{2}-3 x_{3}} & {=0} \\ {-9 x_{1}+2 x_{2}+5 x_{3}} & {=1} & {-9 x_{1}+2 x_{2}+5 x_{3}} & {=5} \\ {4 x_{1}+x_{2}-6 x_{3}} & {=9} & {4 x_{1}+x_{2}-6 x_{3}} & {=45}\end{array}
$$
It can be shown that the first system has a solution. Use this fact and the theory from this section to explain why the second system must also have a solution. Make no row operations.)

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Key Concepts

Consistency of Linear Systems

A linear system is said to be consistent if there is at least one solution that satisfies all its equations simultaneously. In the context of these problems, if one version of the system has been demonstrated to have a solution, then that system is consistent, serving as a baseline for comparing modifications to the equations.

Equivalence Under Nonzero Scalar Multiplication

Multiplying an equation by a nonzero scalar produces a new equation that is equivalent to the original, meaning that both equations have precisely the same solution set. Since in the second system the right-hand sides of some equations are the result of multiplying the original right-hand sides by 5 (a nonzero scalar), the structure of those equations remains equivalent, preserving the solution set from the first system.

Preservation of Solutions in Equivalent Systems

When a system of linear equations is transformed by operations (such as multiplying by nonzero scalars) that are guaranteed to be reversible, the solution set is preserved. Therefore, if the first system is known to be consistent, then any system whose equations are obtained by such transformations—like the second system—is also guaranteed to have a solution.

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