Question 296952
1.) 4x + 3y = 10
2.) 12x + 9y = 28 
a) inconsistent - no solutions
b) independent - one solution
c) dependent - many solution 

If we multiply both sides of equation 1.) by 3 we then have:


1.) 12x + 9y = 30 and
2.) 12x + 9y = 28

The left sides are the same but the right sides are different so there are no values for x and y that will satisfy both equations simultaneously (i.e. no solutions).

For the second pair of equations: 
3.) 3x + 2y = 7
4.) -4x - 3y = 10 

If we rewrite these two equations in slope-intercept form (i.e. y = m*x + b where m is the slope of the line) we have:

3.) 2y = -3*x + 7
     y = (-3/2)*x + 7/2 
4.) 3y = -4*x - 10
     y = (-4/3)*x - 10/3

These two lines have different slopes (-3/2 and -4/3) so are not parallel and will intersect in a single point (i.e. one solution).