91Ó°ÊÓ

An ordered pair consists of two specific numbers written in a particular order, usually represented as \( (x, y) \). The first number corresponds to the horizontal position (x-value) and the second number to thevertical position (y-value) on a graph. Ordered pairs are essential in the representation of equations and functions in a two-dimensional coordinate system.

For instance, in the ordered pair \( \left( \frac{5}{6}, 7 \right) \), the value \( \frac{5}{6} \) is the x-value, and7 is the y-value. These values are used to test if the pair satisfies an equation when substituted for x and y.Solving exercises with ordered pairs helps us understand how changes in x and y influence the outcome of anequation. This is fundamental because ordered pairs can represent solutions to equations, points on a graph, and evenc can help in visualizing data and understanding geometrical shapes.

The substitution method is a strategy used for solving systems of equations or verifying solutions of equations. It involves substituting the values of one variable from an ordered pair into a given equation to determine whether the equation holds true.

In our exercise, we substituted the values from the ordered pair \( \left( \frac{5}{6}, 7 \right) \) into the equation\( y - 6x = 12 \). Given \( x = \frac{5}{6} \) and \( y = 7 \), we replaced these variables:

• Substitute \( x = \frac{5}{6} \) into the equation: \( 6 \times \frac{5}{6} \) simplifies to 5.
• Substitute \( y = 7 \) into the equation and compute \( 7 - 5 \) which equals 2.

This method helps confirm whether the selected values (from the ordered pair) satisfy the equation by ensuring theleft-hand side equals the right-hand side. If they do not equal, as shown in our example, the ordered pair isnot a solution to the equation.

Equation solving involves finding the values for variables that make a given equation true. The process can include substitution, simplification, and comparison to determine if an ordered pair can be a solution.

Our starting point is usually an equation, such as \( y - 6x = 12 \). When solving it with a specific ordered pairlike \( \left( \frac{5}{6}, 7 \right) \), certain steps are followed:

• Substitute the x and y values from the ordered pair into the equation.
• Simplify the expression to compute the left side of the equation.
• Compare the result with the right side of the equation to check if they are equal.

In our exercise: we substituted our ordered pair and arrived at \( 7 - 5 = 2 \), which did not equal 12 (theright-hand side of the equation). This indicates that the ordered pair does not satisfy the equation. Thus, mastering these steps is crucial as it allows one to determine the validity of solutions in mathematical problems and verify theaccuracy of computations in practical applications.