Which of the lines in the figure above contains only points (x,y) with x = y?
Show Hint
Remember the graphs of two fundamental lines: \(y=x\) and \(y=-x\). The line \(y=x\) has a positive slope and goes through quadrants I and III. The line \(y=-x\) has a negative slope and goes through quadrants II and IV. Recognizing these instantly can save time.
Step 1: Understanding the Concept:
This question asks us to identify the graph of the linear equation \(x=y\) (which is the same as \(y=x\)) from a set of plotted lines. Step 2: Detailed Explanation:
The equation \(y=x\) describes a line where the y-coordinate of every point is equal to its x-coordinate. Key characteristics of this line are:
It passes through the origin (0,0), because if \(x=0\), then \(y=0\).
It has a slope of 1. For every 1 unit you move to the right on the x-axis, you also move 1 unit up on the y-axis.
It passes through all points where the coordinates are identical, such as (1,1), (2,2), (-1,-1), (-3,-3), etc.
It bisects the first and third quadrants.
Now let's examine the lines in the figure:
Line A: Passes through the second and fourth quadrants. It has a negative slope. Points on this line might be (-2,2) and (2,-2), which satisfy \(y=-x\).
Line B: Passes through the origin (0,0) and bisects the first and third quadrants. It clearly goes through points like (1,1), (2,2), etc., based on the grid. This is the line \(y=x\).
Line C: Passes through the y-axis at a negative value and has a positive slope. It does not pass through the origin.
Line D: Passes through the y-axis at a positive value and has a positive slope. It does not pass through the origin.
Line E: A horizontal line passing through the y-axis at a negative value. This has the form \(y=k\) where k is a negative constant.
Step 3: Final Answer:
Line B is the only line that contains only points where \(x=y\).
