graph each system of equations and describe it as consistent and independent, consistent and depedneent, or inconsistent
40. x-6y=12
3x+18y=14
41. -5x-6y=13
12y+10x=-26
42. 8y-3x=15
-16y+6x=-30
To graph each system of equations, we can rewrite the equations in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
40. x - 6y = 12
Rearrange the equation: x = 6y + 12
In slope-intercept form: y = (1/6)x - 2
3x + 18y = 14
Rearrange the equation: 18y = -3x + 14
Divide both sides by 18: y = (-1/6)x + 7/9
The graph of the system will have the lines y = (1/6)x - 2 and y = (-1/6)x + 7/9.
To determine if the system is consistent and independent, consistent and dependent, or inconsistent, we can analyze the lines:
The slopes of the lines are different (1/6 and -1/6), and they have different y-intercepts (-2 and 7/9). Therefore, the lines intersect at a unique point.
Hence, the system of equations is consistent and independent.
41. -5x - 6y = 13
Rearrange the equation: y = (-5/6)x - (13/6)
12y + 10x = -26
Rearrange the equation: y = - (5/6)x - (13/12)
The graph of the system will have the lines y = (-5/6)x - (13/6) and y = - (5/6)x - (13/12).
The slopes of the lines are the same (-5/6), and they have different y-intercepts (-13/6 and -13/12). Therefore, the lines are parallel and do not intersect.
Hence, the system of equations is inconsistent.
42. 8y - 3x = 15
Rearrange the equation: y = (3/8)x + 15/8
-16y + 6x = -30
Rearrange the equation: y = (3/2)x + 15/8
The graph of the system will have the lines y = (3/8)x + 15/8 and y = (3/2)x + 15/8.
The slopes of the lines are different (3/8 and 3/2), but they have the same y-intercept (15/8). Therefore, the lines are different but intersect at one point.
Hence, the system of equations is consistent and independent.
To determine whether each system of equations is consistent and independent, consistent and dependent, or inconsistent, we can graph the equations and analyze the resulting graph.
For equation 40:
x - 6y = 12
3x + 18y = 14
To graph these equations, we can rearrange them into slope-intercept form (y = mx + b):
For the first equation:
x - 6y = 12
-6y = -x + 12
y = (1/6)x - 2
For the second equation:
3x + 18y = 14
18y = -3x + 14
y = (-1/6)x + 14/18
y = (-1/6)x + 7/9
Now, let's plot these lines on a graph:
The first equation, y = (1/6)x - 2, has a y-intercept of -2 and a slope of 1/6. So, we can plot one point at (0, -2) and use the slope to get another point. For example, moving 6 units to the right and 1 unit up from the y-intercept, we get the point (6, -1).
The second equation, y = (-1/6)x + 7/9, has a y-intercept of 7/9 and a slope of -1/6. We can plot one point at (0, 7/9) and use the slope to get another point. For example, moving 6 units to the left and 1 unit down from the y-intercept, we get the point (-6, 4/9).
When we plot these two lines on a graph, it looks like the lines intersect at a single point. This means the system is consistent and independent since it has a unique solution.
For equation 41:
-5x - 6y = 13
12y + 10x = -26
Following the same process, we can rearrange the equations into slope-intercept form:
For the first equation:
-5x - 6y = 13
-6y = 5x + 13
y = (-5/6)x - 13/6
For the second equation:
12y + 10x = -26
12y = -10x - 26
y = (-10/12)x - 26/12
y = (-5/6)x - 13/6
Both equations have the same slope (-5/6) and the same y-intercept (-13/6). This means the lines are coincident, and the system is consistent and dependent. There are infinite solutions, and all points on the line represent a solution.
For equation 42:
8y - 3x = 15
-16y + 6x = -30
Rearranging the equations into slope-intercept form:
For the first equation:
8y - 3x = 15
8y = 3x + 15
y = (3/8)x + 15/8
For the second equation:
-16y + 6x = -30
-16y = -6x - 30
y = (6/16)x + 15/8
y = (3/8)x + 15/8
Again, both equations have the same slope (3/8) and the same y-intercept (15/8). This means the lines are coincident, and the system is consistent and dependent. There are infinite solutions, and all points on the line represent a solution.
In summary:
40. The system of equations is consistent and independent.
41. The system of equations is consistent and dependent.
42. The system of equations is consistent and dependent.