Problem 16 - Prealgebra

Question

For each equation, complete the given ordered pairs. $$y=\frac{1}{3} x \quad(-3,),(0,),(3,)$$

Step-by-Step Solution

Answer

The completed ordered pairs are (-3, -1), (0, 0), and (3, 1).

Step 1: Understanding the Equation

We are given the linear equation \( y = \frac{1}{3} x \). This equation represents a straight line. To find the \( y \)-value for a given \( x \)-value, substitute \( x \) into the equation.

Step 1: Complete the Ordered Pair (-3, )

Substitute \( x = -3 \) into the equation: \( y = \frac{1}{3}(-3) = -1 \). So the ordered pair becomes \((-3, -1)\).

Step 2: Complete the Ordered Pair (0, )

Substitute \( x = 0 \) into the equation: \( y = \frac{1}{3}(0) = 0 \). So the ordered pair becomes \((0, 0)\).

Step 3: Complete the Ordered Pair (3, )

Substitute \( x = 3 \) into the equation: \( y = \frac{1}{3}(3) = 1 \). So the ordered pair becomes \((3, 1)\).

Key Concepts

Ordered Pairs

An ordered pair represents a pair of numbers which indicate the position of a point on a Cartesian plane. The format of an ordered pair is generally written as \((x, y)\), where \(x\) represents the horizontal coordinate and \(y\) represents the vertical coordinate.
An ordered pair is crucial when working with linear equations because it helps to define particular points on the graph of the equation.

• The first element \(x\), known as the "abscissa," denotes distance from the vertical axis.
• The second element \(y\), known as the "ordinate," denotes distance from the horizontal axis.

To resolve ordered pairs from a linear equation like \(y = \frac{1}{3}x\), you substitute the given \(x\) values into the equation to solve for \(y\). This gives the completed ordered pairs which can then be plotted on a graph to visualize the line.

Substitution Method

The substitution method is a straightforward approach to solving equations by substituting known values into the equation to find unknown ones. In the context of linear equations, this involves replacing the variable \(x\) in the equation with a specific numerical value to calculate the corresponding \(y\).

• Begin by checking what \(x\) values are provided. In our example \(-3\), \(0\), and \(3\) were given.
• Substitute each \(x\) value into the equation individually. For instance, when \(x = -3\), putting this into \(y = \frac{1}{3}x\) yields \(y = -1\).

The substitution method is particularly helpful when you are tasked with determining specific coordinates on a graph. Practicing this method enhances understanding of how linear equations behave for different inputs.

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that involves drawing the line represented by an equation on a Cartesian plane. Visualizing linear equations helps students understand relationships between variables and interpret data more effectively.
To graph a linear equation based on our example \(y = \frac{1}{3} x\):

• Calculate a series of ordered pairs using the substitution method. For the values given, these would be \((-3, -1)\), \((0, 0)\), and \((3, 1)\).
• Plot these points on a Cartesian plane. Locate each ordered pair by determining the intersection of the \(x\) and \(y\) coordinates on the grid.
• Draw a straight line through all the plotted points. This line represents all solutions to the equation \(y = \frac{1}{3}x\).

Seeing the plotted line visually clarifies the linear relationship, demonstrating the constant rate of change represented in the equation.