if the slope is the same and the y-intercept is the same, then the equations form an identical line and have an infinite number of solutions.
if the slope is the same and the y-intercept is different, then the equations form parallel lines and have no solutions.
otherwise, the equations will form lines that intersect at one common point on a two dimensional graphing plane.
since your equations are already in slope intercept form, then:
first equation is y = -3x + 7 and second equation is y = -3x + 7 leads to a common line which has an infinite number of solutions, i.e. all points on both lines are common to the other.
in standard form, they may not look like they're identical.
in fact, they will probably be multiples of each other.
consider:
5x + 7y = 14
10x + 14y = 28
divide the second equation by 2 to get 5x + 7y = 14.
first and second equation are the same, therefore will generate one common line where all points on the first line are common to all points on the second line.
first equation is y = -3x + 7 and second equation is y = -3x + 14
slope is the same but y-intercept is different.
these equations are are parallel and will have no solution.
first equation is y = -3x + 7
second equation is y = 5x + 7
these equations are neither parallel or identical, therefore they will have one common point between both of them.
the terminology is as follow:
the terminology from varsity tutors is as follows:
If a system has at least one solution, it is said to be consistent.
If a consistent system has exactly one solution, it is independent.
If a consistent system has an infinite number of solutions, it is dependent.
When you graph the equations, both equations represent the same line.
If a system has no solution, it is said to be inconsistent . The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.
under these definitions.
y = -3x + 7 and y = -3x + 7 is consistent and dependent because it has an infinite number of solutions.
y = -3x + 7 and y = -3x + 14 is inconsistent because it has no solutions.
y = -3x + 7 and y = 5x + 7 is consistent and independent because it has one solution.
the graphs of these equations are shown below:
here's a reference.
https://www.varsitytutors.com/hotmath/hotmath_help/topics/consistent-and-dependent-systems
here's another reference.
https://www.purplemath.com/modules/systlin1.htm