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In mathematics, an ordered pair essentially acts as a point on a graph. It is expressed in the form \((x, y)\). Here, the first value is the "x-coordinate" and the second value is the "y-coordinate".
Ordered pairs are used to identify the position of points in a plane.
To determine if an ordered pair is a solution to a system of linear equations,we substitute these values into each equation. If both equations are satisfied, the ordered pair is a solution to the system.

• An ordered pair like \((3, 4)\) means 3 is the x-value and 4 is the y-value.
• Place these values into the system of equations to check if both sides of each equation remain equal after substitution.

This process checks whether the given point lies on the lines represented by these equations.

Solution verification is the process of checking if a given ordered pair solves a system of equations. This involves substituting the values into each equation and checking if they hold true.
The principle is simple:

• Substitute the x-value and y-value of the pair into each equation one at a time.
• Simplify both sides of each equation.
• See if both sides of every equation match after substitution.If so, the ordered pair is a solution; if not, it is not a solution.

Example: Consider the system:\[\begin{align*}3x - y & = 5\x + 2y & = 11\end{align*}\]For the ordered pair \((3,4)\):- Substitute into the first equation: 3(3) - 4 = 5. Result confirmed.- Substitute into the second equation: 3 + 2(4) = 11. Result confirmed.Thus, \((3,4)\) is a solution. The process is repeated similarly for other pairs like \((0, -5)\). If one equation fails, the pair is not a solution to the system, as seen with \((0, -5)\).

The substitution method is a strategy to solve a system of equations by substituting the value of one variable into another equation. It simplifies a system to a single equation with one unknown, which makes finding the solution straightforward.Here is the general approach:

• Solve one equation for one variable in terms of the other.
• Substitute this expression into the other equation.
• Solve the resulting equation for the remaining variable.
• Substitute back to find the other variable's value.

Let's use this method in the context of verifying solutions:- Consider the same system: \(3x - y = 5\) and \(x + 2y = 11\).- If you assume \(x = 3\), substitute 3 into the equation \(x + 2y = 11\): \[3 + 2y = 11\]- Simplify to find \(y = 4\).- Substitute back to check if \(3x - y = 5\) holds true: \[3(3) - 4 = 5\]After checking, we see \((3,4)\) is a valid solution. This method structures the solution-finding process and ensures each step is verified.