Verify that the values of the variables listed are solutions...

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{array}{l}
\left\{\begin{aligned}
3 x+3 y+2 z &=4 \\
x-3 y+z &=10 \\
5 x-2 y-3 z &=8
\end{aligned}\right. \\
x=2, y=-2, z=2 ;(2,-2,2)
\end{array}
$$

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Key Concepts

System of Linear Equations

A system of linear equations is a collection of equations that involve the same set of variables; the solution is a set of values that satisfies all the equations simultaneously. These systems often represent the intersection of geometric entities, like lines or planes, and solving the system means finding the common point or set of points where all these entities meet.

Solution Verification

Solution verification is the process of confirming that a proposed solution satisfies each equation in a system. This involves substituting the candidate values into each equation and checking if the left-hand side equals the right-hand side in every case, thus affirming that the solution is indeed valid.

Substitution Method

The substitution method involves replacing the variables in an equation with their given numerical values. This technique is useful for directly evaluating whether each equation in the system holds true with the candidate solution, effectively confirming the correctness of the solution if all equations are satisfied.

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