Problem 12 - Prealgebra

Question

Complete the given ordered pairs, and use the results to graph the equation. (GRAPH CANT COPY) $$y=-\frac{1}{2} x \quad(0, \quad),(2, \quad),(-2,)$$

Step-by-Step Solution

Answer

The completed pairs are \((0, 0), (2, -1), (-2, 1)\). Graph the line passing through these points for the equation \( y = -\frac{1}{2}x \).

Step 1: Substitute in the First Pair

Start with the equation \( y = -\frac{1}{2}x \). For the initial ordered pair \((0, \quad)\), substitute \( x = 0 \) into the equation: \[ y = -\frac{1}{2} \times 0 = 0 \]. So, the completed pair is \((0, 0)\).

Step 2: Substitute in the Second Pair

Next, use the equation \( y = -\frac{1}{2}x \) for the ordered pair \((2, \quad)\). Substitute \( x = 2 \): \[ y = -\frac{1}{2} \times 2 = -1 \]. Thus, the completed pair is \((2, -1)\).

Step 3: Substitute in the Third Pair

Finally, for the ordered pair \((-2, \quad)\), substitute \( x = -2 \) into the equation: \[ y = -\frac{1}{2} \times (-2) = 1 \]. The completed pair is \((-2, 1)\).

Step 4: Plot the Points and Graph

Now that you have the completed pairs \((0, 0), (2, -1),\) and \((-2, 1)\), plot these points on a coordinate plane.Draw a line through these points to graph the equation \( y = -\frac{1}{2} x \). The line should show a decreasing trend from left to right.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are fundamental in graphing equations on a coordinate plane. An ordered pair consists of two components: an x-coordinate and a y-coordinate, denoted as \((x, y)\). This pair specifies a unique point on the plane. To truly understand ordered pairs, it's essential to see how these components correspond to movements on the Cartesian coordinate system, which consists of two perpendicular axes:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.

When completing an ordered pair for a given equation, we substitute a known value for one component to solve for the other. For example, if you have an equation like \(y = -\frac{1}{2}x\) and know that \(x = 2\) for one of your pairs, you substitute this into the equation to find the corresponding y-value. This input-output process helps in plotting accurate points on the coordinate plane.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It's an essential tool in geometry and algebra for visualizing relationships between variables. This plane is divided by two axes:

• x-axis: A horizontal line where y is zero.
• y-axis: A vertical line where x is zero.

Each point on this plane is defined by an ordered pair \((x, y)\), as discussed earlier. To graph equations like \(y = -\frac{1}{2}x\), you'll need to plot multiple points that satisfy the equation. Plotting involves:

• Moving right (or left) according to the x-coordinate.
• Moving up (or down) according to the y-coordinate.

Once your points are plotted, drawing a line through them reveals the graph of the equation. The line represents all potential solutions for the equation within the given constraints of x and y.

Exploring the Slope-Intercept Form

The slope-intercept form of a linear equation, \(y = mx + b\), is where graphing often starts. In this format:

• \(m\): Represents the slope (how steep the line is).
• \(b\): Represents the y-intercept (where the line crosses the y-axis).

Understanding this form is helpful because it gives immediate insight into the line's behavior. In our example of \(y = -\frac{1}{2}x\):

• The slope \(m = -\frac{1}{2}\) tells us the line falls as it moves from left to right, indicating a decrease.
• Since there's no additional number added or subtracted from \(x\), the y-intercept \(b = 0\) shows the line crosses the origin \((0, 0)\).

To use the slope-intercept form effectively, beyond just calculating specific ordered pairs, you recognize the overall pattern of the line, which can help in predicting and checking other points along the line.