Problem 18 - Algebra 1: Concepts and Skills

Question

Find three ordered pairs that are solutions of the equation. $$ x=9 $$

Step-by-Step Solution

Answer

The ordered pairs that satisfy the equation x=9 are (9,0), (9,1), and (9,-1).

Step 1: Understand the Given Equation

The given equation is x=9. This is a simple linear equation where the x-value is constant and is always equal to 9. The y-value can be any real number.

Step 2: Choose Values for y

Since there are no constraints on the value of y, select arbitrary values. For the purpose of this exercise, let's select y=0, y=1, and y=-1.

Step 3: Form the Ordered Pairs

Substitute the chosen y-values from Step 2 into the equation to get the ordered pairs. They are (9,0), (9,1), and (9,-1).

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a set of two numbers used to represent a point on a graph. An ordered pair is typically written in the form \((x, y)\), where \(x\) and \(y\) are numerical values that define a specific point in a coordinate system.

Ordered pairs are crucial for plotting points on a Cartesian plane and connecting equations with geometric figures. They help establish a relationship between two variables. For instance, in the equation \(x = 9\), the \(x\)-value remains constant at 9, while the \(y\)-value can vary freely. This means that no matter what \(y\) value you choose, the \(x\)-value in the ordered pair will always be 9.

Here are the steps to form an ordered pair:

• Identify the values of \(x\) and \(y\).
• Pair them as \((x, y)\).
• Plot them on the coordinate plane to visualize.

This systematic way of forming pairs allows us to find solutions to linear equations easily.

Constant Value

A constant value in a mathematical equation means that it remains the same throughout the solution process. In the linear equation \(x = 9\), the \(x\)-value is constant. Regardless of which \(y\) value you select, \(x\) will always be 9.

Constant values simplify the equation, making it clear that one variable doesn't change. This can help you predict the formation of a vertical line on a graph, as all points along the line share the same \(x\)-value.

• In \(x = 9\), \(x\) is always 9.
• The equation doesn't require adjustments for different \(y\) values.
• This setup describes a vertical line parallel to the \(y\)-axis.

Recognizing a constant value is key to interpreting graphs and understanding how different variables relate in an equation.

Equation Solutions

In mathematical terms, a solution to an equation is a set of values that satisfy the equation. For the equation \(x = 9\), solutions are composed of ordered pairs where \(x\) is constant.

Since the equation has \(x = 9\) without any restriction on \(y\), there are virtually unlimited solutions. Choosing different \(y\)-values does not impact the \(x\)-value. Thus, you can easily form solutions by deciding on \(y\) values:

• Select any real number for \(y\).
• Pair it with the constant \(x = 9\).
• Examples include \((9, 0)\), \((9, 1)\), and \((9, -1)\).

Understanding how solutions are formed allows us to conceptualize what kinds of graphs these equations produce, revealing insights about their nature and behavior.