graph each system of equations and describe it as consistent and independent, consistent and depedneent, or inconsistent

37. y=3x-4
y=6x-8

38. y=2x-1
y=2x+6

39. 2x+5y=10
-4x-10y=20

To graph each system of equations and determine its consistency, we can start by rearranging the equations in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.

37.
The given equations are:
y = 3x - 4
y = 6x - 8

To graph the system, we can proceed as follows:
1. Take the first equation: y = 3x - 4
- Plot the y-intercept at -4 (when x = 0, y = -4)
- From the y-intercept, use the slope 3 to find another point. The slope of 3 means that for every increase of 1 in x, y increases by 3. So when x = 1, y = -1 (-4 + 3), and when x = 2, y = 2 (-4 + 3 + 3).
- Draw a line passing through the two plotted points.

2. Take the second equation: y = 6x - 8
- Plot the y-intercept at -8 (when x = 0, y = -8)
- From the y-intercept, use the slope 6 to find another point. The slope of 6 means that for every increase of 1 in x, y increases by 6. So when x = 1, y = -2 (-8 + 6), and when x = 2, y = 4 (-8 + 6 + 6).
- Draw a line passing through the two plotted points.

Now, analyze the graph:
Since both lines have different slopes (3 and 6), they are not parallel, nor do they coincide, leading us to an intersection point. Therefore, the system is consistent and independent.

38.
The given equations are:
y = 2x - 1
y = 2x + 6

To graph the system, we can follow these steps:
1. Take the first equation: y = 2x - 1
- Plot the y-intercept at -1 (when x = 0, y = -1)
- From the y-intercept, use the slope 2 to find another point. The slope of 2 means that for every increase of 1 in x, y increases by 2. So when x = 1, y = 1 (-1 + 2), and when x = 2, y = 3 (-1 + 2 + 2).
- Draw a line passing through the two plotted points.

2. Take the second equation: y = 2x + 6
- Plot the y-intercept at 6 (when x = 0, y = 6)
- From the y-intercept, use the slope 2 to find another point. The slope of 2 means that for every increase of 1 in x, y increases by 2. So when x = 1, y = 8 (6 + 2), and when x = 2, y = 10 (6 + 2 + 2).
- Draw a line passing through the two plotted points.

Now, analyze the graph:
Both lines have the same slope (2) and different y-intercepts (-1 and 6). They do not coincide or intersect. Therefore, the system is inconsistent or has no solution, which means it is inconsistent and independent.

39.
The given equations are:
2x + 5y = 10
-4x - 10y = 20

To graph the system, we'll rewrite the equations in slope-intercept form:
1. Take the first equation: 2x + 5y = 10
- Solve for y: y = (-2/5)x + 2
- Plot the y-intercept at 2 (when x = 0, y = 2)
- Use the slope (-2/5). For an increase of 1 in x, y decreases by 2/5. So when x = 1, y = 8/5 (2 - 2/5), and when x = 2, y = 6/5 (2 - 2/5 - 2/5).
- Draw a line passing through the two plotted points.

2. Take the second equation: -4x - 10y = 20
- Solve for y: y = (-2/5)x - 2
- Plot the y-intercept at -2 (when x = 0, y = -2)
- Use the slope (-2/5). For an increase of 1 in x, y decreases by 2/5. So when x = 1, y = -12/5 (-2 - 2/5), and when x = 2, y = -22/5 (-2 - 2/5 - 2/5).
- Draw a line passing through the two plotted points.

Now, analyze the graph:
Both lines have the same slope (-2/5) but different y-intercepts (2 and -2). They are parallel and do not intersect. Therefore, the system is consistent and dependent, meaning it has infinitely many solutions since the lines coincide.