Problem 66 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the line whose points have equal \(x\) - and \(y\) -coordinates. What would the equation of this line be?

Step-by-Step Solution

Answer

The equation of the line is x = y.

Step 1 - Understanding the Problem

To sketch the graph of the line where the points have equal x- and y-coordinates, we need to find points where x equals y.

Step 2 - Identify Points on the Line

Choose several points where the coordinate pair is the same. Example points could be (1,1), (2,2), (3,3), (-1,-1), etc. These points satisfy the condition x = y.

Step 3 - Plot the Points on the Coordinate Plane

Plot the points (1,1), (2,2), (3,3), (-1,-1), etc. on the coordinate plane.

Step 4 - Draw the Line

Draw a straight line through the plotted points. This line should pass through all these points because on this line, the x-coordinate will always equal the y-coordinate.

Step 5 - Determine the Equation of the Line

Since the x-coordinate equals the y-coordinate for every point on the line, the equation of the line is simply x = y.

Key Concepts

Coordinate Plane

The coordinate plane is a key concept in graphing linear equations. It is a two-dimensional surface defined by two perpendicular axes. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. The point where these axes intersect is called the origin, represented by the coordinates (0,0). Each point on the coordinate plane can be described by an ordered pair \((x, y)\). For example, the point (2,3) is 2 units to the right of the origin and 3 units up. Understanding the coordinate plane is essential for plotting points and graphing lines.

Plotting Points

Plotting points is the fundamental step in graphing linear equations. To plot a point, you need its coordinates \((x, y)\). Start at the origin (0,0). Move horizontally to the x-coordinate: right if x is positive, left if x is negative. Then move vertically to the y-coordinate: up if y is positive, down if y is negative. Mark the point where the two steps meet. For example, to plot (3,3), move 3 units to the right along the x-axis and then 3 units up the y-axis. Repeat this process for each point you need to plot. Once all points are placed, you can see the pattern or line they form. This step is crucial for visualizing equations and finding lines.

Line Equation

A line equation represents the relationship between x and y coordinates on a graph. The simplest form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept is where the line crosses the y-axis. In our example, we have a very special line equation: x = y. This means the x- and y-coordinates are always equal. Start by choosing a few points like (1,1), (2,2), and (-1,-1). Plot these points and notice they all form a straight line. This line goes through the origin and forms a 45-degree angle with both axes. The equation x = y simplifies the process, making it clear that any point on this line will have equal x and y values.