91Ó°ÊÓ

An ordered pair is a fundamental concept in mathematics, especially in coordinate geometry, where it is used to describe the location of a point on a plane.

• The first element of an ordered pair represents the x-coordinate (horizontal position).
• The second element represents the y-coordinate (vertical position).

For example, in the ordered pair \((-4, -51)\),

• -4 is the x-coordinate, and
• -51 is the y-coordinate.

Understanding ordered pairs is crucial for plotting points on a graph and visualizing solutions to equations. This basic structure allows us to link numerical relationships such as those found in linear equations like \(y = 12x - 3\). Given specific x-values, substituting these into the equation will yield a unique y-value, therefore forming an ordered pair.

The substitution method is a powerful tool used to solve equations, particularly in finding solutions for a linear equation. Here's how it works:

• One variable in an equation is replaced with a value or another expression.
• This action creates a simpler equation to solve.

In the exercise, the substitution method is applied to find the solutions to the equation \(y = 12x - 3\) using given x-values. For instance:

• Substitute \(x = -4\) into the equation: \(y = 12(-4) - 3\) and solve for \(y\).
• Perform similar substitutions for \(x = 0\) and \(x = 4\).

This technique helps us verify the connection between variables and underscores the principle that an equation describes a whole set of ordered pairs - solutions that can be plotted on a graph.

Coordinate geometry, also known as analytic geometry, is a method of representing and analyzing geometric figures using algebraic equations. It links algebra and geometry to visualize concepts by plotting points, lines, and curves.Using the Cartesian Coordinate System:

• The horizontal axis is the x-axis,
• The vertical axis is the y-axis.

Points are represented as ordered pairs \((x, y)\) in this system. In the example we worked with, we calculate different ordered pairs such as \((-4, -51)\), \((0, -3)\), and \((4, 45)\).By plotting these points on a graph, we can visualize the linear equation \(y = 12x - 3\) as a straight line. This helps in understanding the equation's properties - such as its slope and intercepts, offering a visual representation of algebraic relationships.