When plotted on a coordinate grid, a sidewalk in a park can be modeled with the equation 2x-y=7 .

Question

Complete each sentence below by dragging and dropping the correct equations into the boxes.



The owner of the park plans to build two straight walkways parallel to the first walkway. The equations of these two walkways COULD be
and
.
The owner also plans to build a straight walkway perpendicular to the first walkway. The equation of this walkway COULD be
.

Answer 1

To solve this problem, we need to determine the equations of lines that are parallel and perpendicular to the given equation of the sidewalk, which is \(2x - y = 7\).

1. Finding the slope of the original line: We can rewrite the equation in slope-intercept form \(y = mx + b\): \[ y = 2x - 7 \] From this, we can see that the slope (\(m\)) of the original line is \(2\).

2. Finding the equations of lines parallel to the original line: Parallel lines have the same slope. Therefore, any line parallel to \(2x - y = 7\) will also have a slope of \(2\). These lines can be represented in the form: \[ y = 2x + b \] where \(b\) can be any real number.

   Some possible parallel line equations could be:

   • \(y = 2x + 1\)
   • \(y = 2x - 3\)

3. Finding the equation of a line perpendicular to the original line: Perpendicular lines have slopes that are negative reciprocals of each other. The slope of the original line is \(2\), so the slope of a perpendicular line would be \(-\frac{1}{2}\). The equation of a perpendicular line can be represented in the form: \[ y = -\frac{1}{2}x + b \] where \(b\) can again be any real number.

   A possible perpendicular line equation could be:

   • \(y = -\frac{1}{2}x + 3\)

Answers:

• The equations of the two walkways parallel to the first walkway COULD be:

  • \(y = 2x + 1\)
  • \(y = 2x - 3\)

• The equation of the walkway perpendicular to the first walkway COULD be:

  • \(y = -\frac{1}{2}x + 3\)

You can replace the above equations with any other equations meeting the criteria mentioned.