Problem 2 - Prealgebra

Question

Complete the given ordered pairs, and use the results to graph the equation. (GRAPH CANT COPY) $$x-y=2 \quad(1, \quad),(2,),(, 2)$$

Step-by-Step Solution

Answer

The completed ordered pairs are (1, -1), (2, 0), and (4, 2).

Step 1: Understand the Equation

The given equation is \( x - y = 2 \). This represents a linear equation that can be used to plot a line on a graph.

Step 2: Solve for Missing Values in Ordered Pairs

To complete the ordered pairs, substitute the provided values into the equation. We have the partial pairs: \((1, \, ?), \, (2, \, ?), \, (?, \, 2)\).

Step 3: Complete First Ordered Pair \((1, \?)\)

Substitute \( x = 1 \) into the equation: \[ \begin{align*} 1 - y &= 2\ -y &= 1\ y &= -1 \end{align*} \]The completed pair is \((1, -1)\).

Step 4: Complete Second Ordered Pair \((2, \?)\)

Substitute \( x = 2 \) into the equation: \[ \begin{align*} 2 - y &= 2\ -y &= 0\ y &= 0 \end{align*} \]The completed pair is \((2, 0)\).

Step 5: Complete Third Ordered Pair \((?, 2)\)

Substitute \( y = 2 \) into the equation: \[ \begin{align*} x - 2 &= 2\ x &= 4 \end{align*} \]The completed pair is \((4, 2)\).

Step 6: Graph the Equation Using Ordered Pairs

The completed ordered pairs \((1, -1), (2, 0), (4, 2)\) can be plotted on a graph. These points can be connected to show the line represented by the equation \( x - y = 2 \).

Key Concepts

Understanding Ordered Pairs

Ordered pairs are fundamental to graphing equations on a two-dimensional plane. These pairs, \((x, y)\), represent coordinates that correspond to specific points on a graph. The first element in the ordered pair is the x-coordinate and the second is the y-coordinate. Understanding ordered pairs is crucial for visualizing mathematical relationships, such as lines and curves, in a coordinate system.
To use ordered pairs effectively:

• Identify the x-coordinate, which indicates movement along the horizontal axis.
• Identify the y-coordinate, which specifies movement along the vertical axis.
• Accurately plot each pair on the graph to represent the position of a point.

For the given exercise, the ordered pairs were completed by substituting known values into the equation \(x - y = 2\). This provided us with precise locations to plot the line on a graph.

Graphing Linear Equations

Graphing linear equations involves plotting points derived from equations to create straight lines. Linear equations often take the form \(ax + by = c\), and their graphical representation forms a line. It is especially useful for visualizing solutions or patterns.
Here is how you can graph such equations:

• Rewrite the equation in a way that makes it easy to plot, like the slope-intercept form \(y = mx + c\), if comfortable.
• Find two or more ordered pairs by substituting different values for \(x\) or \(y\) (as we did in the exercise).
• Plot these ordered pairs on a coordinate plane.
• Connect the points with a straight line, extending it across the graph.

The resulting graph provides a visual understanding of the equation's solutions. In the exercise, once we had the ordered pairs \((1, -1)\), \((2, 0)\), and \((4, 2)\), we plotted these on a graph to clearly illustrate the linear relationship defined by \(x - y = 2\).

Solving Equations to Determine Coordinates

Solving equations is a critical skill needed to complete ordered pairs, especially in the context of graphing. The goal is to find unknown values by isolating variables using mathematical operations. This process often involves:

• Identifying given values to substitute into the equation.
• Applying arithmetic operations to isolate the variable of interest.
• Ensuring all operations are balanced on both sides of the equation.

In our exercise, to complete each ordered pair, we solved for the missing coordinate by plugging given values into the linear equation \(x - y = 2\). For example, substituting \(x = 1\) allowed us to solve for \(y\), resulting in the pair \((1, -1)\). This technique helps elucidate how each specific point relates to the equation being graphed. By understanding how to solve these equations effectively, students can better interpret and graph linear relationships.