SOLUTION: How many solutions does the system of equations have? 3x+12y=20 y=-1/4x+5/3

To clarify, I am rewriting your equations so it is clear:

3x + 12y = 20
y = -(1/4)x + (5/3)

These equations both represent lines. And with these lines there are 3 possibilities:

a. 0 solutions (lines are parallel, do not cross)
b. 1 solution (lines cross at one x,y point)
c. infinite solutions (lines are identical, laying on top of each other)

...but how do we figure out which it is? We examine them by setting them equal to each other to compare.

First, we simplify the equations so that one of the variables is common in both. Since one equation is already "y=", we will rearrange the other to match:

3x + 12y = 20 can be changed to:
(3/12)x + (12/12)y = (20/12)   [whole equation divided by 12]
(1/4)x + y = (5/3)                   [simplifying the fractions]
y = -(1/4)x + (5/3)                  [moving (1/4)x to the other side]

and as you can see, this matches the 2nd equation of y = -(1/4)x + (5/3).

Since the two equations are identical, there are INFINITE SOLUTIONS. This is because you solve to 0=0 when making y=y:

-(1/4)x + (5/3) = -(1/4)x + (5/3)
-(1/4)x + (1/4)x = (5/3) - (5/3)
0=0


BUT

even if they were not identical, you would still set them equal to each other and simplify as far as you could.

AND

if doing so eliminates all the variables but are left with just numbers (for example 0=3), you would have parallel lines with no solutions.

OR

if doing so solves to a variable solution (for example x=-3), then you have one solution and you would put the x solution into either equation to get your y.


Hope that helps :)

Solve both equations for y, thereby placing them in slope-intercept form. Then inspect them.



3x + 12y = 20_____12y = - 3x + 20__________

 ------ Already in slope-intercept form

As seen, the equations are IDENTICAL, thereby leading to an INFINITE NUMBER of solutions.