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Understanding how to solve systems of linear equations is an essential skill in algebra. In these systems, each equation represents a line, and the solution consists of the point or points where the lines intersect. Solutions can be a single point where all lines intersect, no point if lines are parallel and don't intersect, or an infinite number of points if the lines coincide.

In the given exercise, we're faced with a system of three equations involving three variables: x, y, and z. The objective is to find the values of these variables that satisfy all three equations simultaneously. The step by step solution walks through this process by first rewriting the equations in standard form, isolating one variable, substituting this into other equations, and finally, finding a common solution. By following such structured steps, students can methodically approach complex systems and unravel the solution with clarity and precision.

To solve systems of equations, various algebraic methods can be employed. The two most common techniques are substitution and elimination. Both methods leverage the properties of equality and the ability to manipulate equations to simplify the system to find the values of the unknown variables. The substitution method involves expressing one variable in terms of the others and substituting this expression into the remaining equations. The elimination method focuses on adding or subtracting equations to cancel out one of the variables, making it easier to find the values of the remaining variables.

Understanding when and how to apply these methods is crucial. Factors such as the complexity of the system and the coefficients of the variables can influence the choice of method. The choice becomes clearer with practice, enabling students to select the most efficient strategy to reach the solution.

The substitution method is a strategic way to solve a system of equations where one equation is solved for one variable, and this expression is then substituted into the other equations. This creates a new, simpler system with fewer variables to solve. It's especially effective when one of the equations can easily be solved for one variable.

In our exercise, Step 2 illustrates this by isolating x in the first equation. Substitution is employed again in Step 3 where x is eliminated from two of the equations. This manipulation should be done with care to avoid errors in simplifying and calculation. By systematically substituting and simplifying, we see the system narrowed down to a single variable, allowing for a straightforward solution.

The elimination method solves a system by adding or subtracting equations to eliminate one of the variables. It's optimal when the system's equations can be easily manipulated so that adding them causes one variable to cancel out. This requires attention to the coefficients of the variables: they must be opposites of each other for elimination to work effectively.

In our example, the elimination method is used after the substitution step simplifies the system down to two equations. Step 4 involves multiplying equations strategically so that terms cancel when they are added together. This directly yields the value of y, which can then be used to find z and subsequently x. The elimination method is highly useful when equations are already in a form that makes cancelling straightforward or after substitution has reduced the system's complexity.