Problem 44 - Prealgebra

Question

Graph each ordered pair on a coordinate system. $$Z\left(3 \frac{1}{5}, 3 \frac{1}{4}\right)$$

Step-by-Step Solution

Answer

Plot point \( Z \) at approximately \( x = 3.2 \) and \( y = 3.25 \) on the coordinate plane.

Step 1: Understand the Ordered Pair

The ordered pair given is \( Z\left(3 \frac{1}{5}, 3 \frac{1}{4}\right) \). This indicates a point on a coordinate plane where \( x = 3 \frac{1}{5} \) and \( y = 3 \frac{1}{4} \).

Step 2: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers to improper fractions for easier plotting. For \( x = 3 \frac{1}{5} \), convert it to \( x = \frac{16}{5} \). For \( y = 3 \frac{1}{4} \), convert it to \( y = \frac{13}{4} \).

Step 3: Locate the x-coordinate

On the horizontal axis, plot the point corresponding to the \( x \) value \( \frac{16}{5} \), which is approximately 3.2. Look for this point on the x-axis.

Step 4: Locate the y-coordinate

On the vertical axis, plot the point corresponding to the \( y \) value \( \frac{13}{4} \), which is approximately 3.25. Look for this point on the y-axis.

Step 5: Plot the Point Z

Find the intersection of the \( x \) value (approximately 3.2) and the \( y \) value (approximately 3.25) on the coordinate plane. This intersection is point \( Z \).

Key Concepts

Ordered Pair

An ordered pair consists of two values enclosed in parentheses, representing a specific point on a coordinate plane. This pair follows the format \((x, y)\), where the first number corresponds to the x-axis value and the second number to the y-axis value.
Ordered pairs are crucial in identifying positions on a grid. They help in graphing and spatial analysis.

• The order of the numbers matters; swapping them changes the location on the plane.
• Examples of ordered pairs are \((3, 4)\) or \((0, -2)\).

Understanding ordered pairs lays the groundwork for more complex geometric and algebraic concepts.

Improper Fractions

Improper fractions are fractions where the numerator (top number) is larger than or equal to the denominator (bottom number). For example, \(\frac{7}{5}\) and \(\frac{13}{4}\) are improper fractions.
Converting mixed numbers into improper fractions is often necessary in graphing or calculations because it simplifies the computation process.

• A mixed number like \(3 \frac{1}{5}\) can be converted to an improper fraction by multiplying the whole number by the denominator and adding the numerator: \(3 \times 5 + 1 = 16\), so \(3 \frac{1}{5} = \frac{16}{5}\).
• These representations facilitate easy math operations and plotting on graphs.

become familiar with improper fractions as they are common in many math-related tasks.

Plotting Points

Plotting points on a coordinate plane involves pinpointing the location defined by an ordered pair. This task requires identifying the precise intersection on the graph where both coordinates meet.
Plotting is essential for graphing functions, analyzing data, and drawing geometric shapes.

• Start by locating the x-coordinate on the horizontal axis; then find the y-coordinate on the vertical axis.
• The intersection of these two lines marks the exact location of the point.
• Use graph paper or a digital tool for accurate plotting.

Master plot points to help visualize mathematical problems and solutions.

x-coordinate

The x-coordinate is the first value in an ordered pair and determines the horizontal position on the coordinate plane. In the ordered pair \( (3 \frac{1}{5}, 3 \frac{1}{4}) \), the x-coordinate is \(3 \frac{1}{5}\), which can be expressed as \(\frac{16}{5}\).
The x-coordinate tells how far left or right to move from the origin (0,0).

• Positive x-coordinates mean moving right; negative x-coordinates mean moving left.
• In our example, \(\frac{16}{5}\) is approximately 3.2, indicating a rightward movement from the origin.

Recognizing and interpreting x-coordinates allow for accurate placement of points.

y-coordinate

The y-coordinate is the second value in an ordered pair and indicates the vertical position of a point on the coordinate plane. For \( (3 \frac{1}{5}, 3 \frac{1}{4}) \), the y-coordinate is \(3 \frac{1}{4}\), or \(\frac{13}{4}\) when converted.
The y-coordinate tells how far up or down to move from the origin.

• Positive y-coordinates imply upward movement; negative y-coordinates imply a downward movement.
• In our scenario, \(\frac{13}{4}\) is about 3.25, pointing to an upward shift from the origin.

Understanding and placing y-coordinates precisely is key to graphing points effectively.