Problem 67 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the line whose points have \(x\) - and \(y\) -coordinates that are negatives of each other. What would the equation of this line be?

Step-by-Step Solution

Answer

The equation of the line is \y = -x\.

Step 1: Identify the Relationship Between Coordinates

Recognize that the given points have the property where the coordinates are negatives of each other, i.e., if a point on the line is \((x, y)\), then \((x, y) = (a, -a)\).

Step 2: Form the Equation of the Line

The equation of the line can be derived from the relationship between the coordinates. Since \((x, y) = (a, -a)\), it follows that \y = -x\.

Step 3: Plot Key Points to Sketch the Line

Choose a few points satisfying \y = -x\ for sketching. Examples include \((1, -1)\), \((-2, 2)\), and \((3, -3)\).

Step 4: Draw the Line on the Graph

Plot the chosen points on the coordinate plane: \((1, -1)\), \((-2, 2)\), and \((3, -3)\). Draw a straight line through these points to represent the equation \y = -x\.

Key Concepts

Coordinate Plane

Understanding the coordinate plane is essential in graphing linear equations. The coordinate plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). The center point where these axes intersect is called the origin, with coordinates (0, 0). The quadrants are numbered counterclockwise, starting from the upper right (Quadrant I).

Here's a quick breakdown of the quadrants and the sign of their coordinates:

• Quadrant I: (+x, +y)
• Quadrant II: (-x, +y)
• Quadrant III: (-x, -y)
• Quadrant IV: (+x, -y)

Understanding this layout helps us place points correctly on the graph when we are plotting points for linear equations.

Equation of a Line

An equation of a line describes all the points that lie on the line. For the given exercise, we need to determine an equation where the x-coordinates and y-coordinates are negatives of each other. Given the relationship \(y = -x\), every y-coordinate is the negative of its corresponding x-coordinate.

This relationship can be translated into the line’s equation:
\[y = -x\]

This linear equation means that for every point on the line, the y-value is always the negative of the x-value. It's a simple yet powerful representation, because once you know one coordinate (x or y), you can easily find the other by taking its negative.

Plotting Points

Plotting points is one of the key steps in graphing any linear equation. Let's take the equation \[y = -x\]\ and choose points to plot.

For example:

• When x = 1, y = -1 → (1, -1)
• When x = -2, y = 2 → (-2, 2)
• When x = 3, y = -3 → (3, -3)

After selecting these points, you plot them on the coordinate plane. Once plotted, these points should align in a straight line if done correctly. Drawing a line through these plotted points gives a visual representation of the equation \(y = -x\). This process of plotting multiple points helps to confirm the accuracy of the graphed line.

Negative Coordinates

Negative coordinates are simply points on the coordinate plane where either the x-value or the y-value (or both) are negative. For example, in the point (-2, 3), the x-coordinate is negative, and in the point (4, -5), the y-coordinate is negative.

When both values are negative, such as (-3, -3), the point is located in Quadrant III. Understand that negative coordinates change the direction in which you move on the graph plane:

• Negative x-coordinates: Move left from the origin.
• Negative y-coordinates: Move down from the origin.

In the given exercise, many points we plot involve negative coordinates, so understanding how to handle them is key. This comprehension ensures accurate point plotting and correct line drawing.