Ask a New Question
Menu

Check all of the choices that are true about this system of linear equations (choose 3):

12x−4y=48

y=−34x+12

(3 points)
Responses

These two lines have different slopes
These two lines have different slopes

There is one solution to this system
There is one solution to this system

These two lines have the same slope, but different y-intercepts
These two lines have the same slope, but different y-intercepts

These two lines are actually the same line written in different forms
These two lines are actually the same line written in different forms

These two lines have a multiple solutions that occur positively and negatively
These two lines have a multiple solutions that occur positively and negatively

These two lines are parallel
These two lines are parallel

These two lines have a solution at the point of intersection
These two lines have a solution at the point of intersection

There is no solution to this system

1 answer

To analyze the given system of linear equations, let's first rewrite both equations in a comparable form and find their slopes and y-intercepts.

  1. The first equation is: \[ 12x - 4y = 48 \] We can rearrange it into slope-intercept form (y = mx + b): \[ -4y = -12x + 48 \ y = 3x - 12 \] Thus, the slope (m) of the first line is 3 and the y-intercept (b) is -12.

  2. The second equation is: \[ y = -\frac{34}{1}x + 12 \] Here, the slope (m) is -34 and the y-intercept (b) is 12.

Analysis of the slopes

  • The slope of the first line is 3.
  • The slope of the second line is -34.

Since the slopes are different, the lines are not parallel.

Summary of statements to check:

  1. These two lines have different slopes. (True)
  2. There is one solution to this system. (This may or may not be true depending on whether they intersect.)
  3. These two lines have the same slope, but different y-intercepts. (False)
  4. These two lines are actually the same line written in different forms. (False)
  5. These two lines have multiple solutions that occur positively and negatively. (False)
  6. These two lines are parallel. (False)
  7. These two lines have a solution at the point of intersection. (True - since they intersect at exactly one point as they are not parallel and not the same line.)
  8. There is no solution to this system. (False)

Selecting true statements:

Based on the analysis, the true statements are:

  1. These two lines have different slopes.
  2. There is one solution to this system.
  3. These two lines have a solution at the point of intersection.

Therefore, the correct choices are:

  • These two lines have different slopes.
  • There is one solution to this system.
  • These two lines have a solution at the point of intersection.