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An ordered pair is a set of two numbers used to represent the position of a point in a coordinate system. It is written as \((x, y)\), where the first number represents the x-coordinate and the second number represents the y-coordinate.

• The x-coordinate describes horizontal movement along the x-axis.
• The y-coordinate indicates vertical movement along the y-axis.

Ordered pairs are essential in solving equations because they help determine the specific solutions that satisfy the equation. In the exercise given, we use the ordered pair \((-2, 1)\) to verify if it solves the linear equation \(3x - 5y = 1\). If substitution results in a true statement, where both sides of the equation are equal, then the ordered pair is a solution.

The substitution method is a simple yet effective technique for solving linear equations. It involves replacing variables with known values to check if an equation holds true. When applying substitution:

• Identify known values, like those given in an ordered pair.
• Insert these values into the equation to test validity.
• Solve the equation to see if both sides are equal.

In our exercise, we substitute \(x = -2\) and \(y = 1\) into the equation \(3x - 5y = 1\) to check if the result matches the right-hand side. After substitution and simplification \[3(-2) - 5(1) = -11\] we find that \(-11\) is not equal to \(1\), thus confirming the ordered pair is not a solution through this method.

Linear equations are foundational in mathematics and describe a straight line when graphed. They typically take the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. The solution to a linear equation is the set of ordered pairs \((x, y)\) that make the equation true.

• The coefficients \(a\) and \(b\) determine the slope and direction of the line.
• The constant \(c\) affects the line's position on the graph.

Understanding linear equations helps in predicting and analyzing relationships between variables in various fields. In our task, checking if \((-2, 1)\) is a solution involves confirming whether substituting these values into the equation yields a true statement. If true, the point lies on the line represented by the equation. Otherwise, it does not, as demonstrated by our substitution from the exercise.