SOLUTION: Please help me solve this equation: identify the system of equations. 4x + 3y = 10 12x + 9y = 28 a) inconsistent - no solutions b) independent

SOLUTION: Please help me solve this equation: identify the system of equations. 4x + 3y = 10 12x + 9y = 28 a) inconsistent - no solutions b) independent - one solution c) depen

Question 296952: Please help me solve this equation:
identify the system of equations.

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

identify the system of equations.
3x + 2y = 7
-4x - 3y = 10
a) inconsistent - no solutions
b) independent - one solution
c) dependent - many solutions

Answer by dabanfield(803) (Show Source): You can put this solution on YOUR website!
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).

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