Which of these systems of linear equations could represent the two lines? Select all that apply.

Question

x = 2 and y = 4
x = 4 + y and y = 2 + x
y = 2x - 2 and y = x - 2
x = 2 + y and y = 4 + x
y = x + 2 and y = 2x

Answer 1

To determine which of the provided systems of linear equations represent lines in a coordinate plane, we need to analyze each pair of equations.

1. x = 2 and y = 4:

   • This represents the vertical line x = 2 and the horizontal line y = 4. They intersect at the point (2, 4).

2. x = 4 + y and y = 2 + x:

   • The first equation can be rearranged to y = x - 4 (since x - 4 = y).
   • The second equation can be rearranged to y = x - 2.
   • These equations represent two lines with different slopes (both have a slope of 1), so they are not parallel and will intersect at a point.

3. y = 2x - 2 and y = x - 2:

   • The first equation has a slope of 2, and the second equation has a slope of 1. Since the slopes are different, these lines will intersect.

4. x = 2 + y and y = 4 + x:

   • Rearranging the first equation gives y = x - 2.
   • The second equation rearranges to y = x - 4. Both lines are parallel because they have the same slope (1) and different y-intercepts, so they do not intersect.

5. y = x + 2 and y = 2x:

   • The first equation has a slope of 1 and the second has a slope of 2. They will intersect since they have different slopes.

From the analysis:

• The pairs that represent intersecting lines (which corresponds to systems of linear equations that can represent two lines) are:
  • x = 2 and y = 4
  • x = 4 + y and y = 2 + x
  • y = 2x - 2 and y = x - 2
  • y = x + 2 and y = 2x

The pairs that do not represent two intersecting lines are:

• x = 2 + y and y = 4 + x (because both pairs of equations lead to parallel lines)

Thus, select all that apply:

• x = 2 and y = 4
• x = 4 + y and y = 2 + x
• y = 2x - 2 and y = x - 2
• y = x + 2 and y = 2x