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When working with systems of linear equations, ordered pairs play a crucial role as potential solutions. In the context of a coordinate plane, an ordered pair \( (x, y) \) indicates a point where 'x' represents the horizontal position and 'y' represents the vertical position. To test if an ordered pair is a solution to a system of equations, each element of the pair is substituted into the equations. If both equations are satisfied—meaning they result in true statements—the ordered pair is a valid solution.
For example, considering the pair \( (-5, 9) \) and the system of equations given in the exercise, we substitute '-5' for every 'x' and '9' for every 'y'. If both resulting expressions are true, then \( (-5, 9) \) is indeed a solution to the system. This process checks the compatibility of the ordered pair with the constraints defined by the equations. It's essential to check every ordered pair against all equations in the system to ensure they satisfy the entire system and not just a single equation.

The substitution method is a powerful algebraic technique used to solve systems of equations. This method involves expressing one variable in terms of the other from one equation and then substituting this expression into the other equation. This way, the system, which originally has two variables, is reduced to a single variable.
This method consists of several steps: first, solve one of the equations for one of the variables; next, replace this variable in the second equation with the expression obtained; then, solve the resulting single-variable equation; and finally, back-substitute the found value into one of the original equations to find the other variable's value. Using this method streamlines finding the ordered pair that represents the solution to the system.
In our step-by-step example, the substitution method isn't directly applied since the ordered pair was given for evaluation. However, if we needed to find the solution from scratch, we could use substitution to isolate one variable and solve the system algebraically.

Finding algebraic solutions to systems of equations is at the heart of many algebra problems. Solutions can generally be expressed as ordered pairs that satisfy all equations simultaneously. The algebraic approach encompasses a variety of methods, including substitution, elimination, and graphical methods, to find these solutions.
When we talk about algebraic solutions in the context of the substitution method, we're referring to the manipulation of equations to isolate and solve for unknowns. After obtaining this solution, it's interpreted in terms of ordered pairs that satisfy the given system. Algebraic solutions provide a precise and practical way of understanding the relationships between variables governed by the equations.
We can analyze algebraic solutions to assess the feasibility of certain scenarios, such as supply and demand in economics, the intersection of lines in geometry, or balancing chemical equations in chemistry. Therefore, the ability to understand and calculate algebraic solutions is a fundamental skill in not just mathematics, but in numerous scientific and real-world applications as well.