Question 935582
<p>To clarify, I am rewriting your equations so it is clear:<br /><br />3x + 12y = 20<br />y = -(1/4)x + (5/3)<br /><br />These equations both represent lines. And with these lines there are 3 possibilities:<br /><br />a. 0 solutions (lines are parallel, do not cross)<br />b. 1 solution (lines cross at one x,y point)<br />c. infinite solutions (lines are identical, laying on top of each other)<br /><br />...but how do we figure out which it is? We examine them by setting them equal to each other to compare.<br /><br />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:<br /><br />3x + 12y = 20 can be changed to:<br />(3/12)x + (12/12)y = (20/12)&nbsp;&nbsp; [whole equation divided by 12]<br />(1/4)x + y = (5/3)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; [simplifying the fractions]<br />y = -(1/4)x + (5/3)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp; [moving (1/4)x to the other side]<br /><br />and as you can see, this matches the 2nd equation of y = -(1/4)x + (5/3).<br /><br />Since the two equations are identical, there are INFINITE SOLUTIONS. This is because you solve to 0=0 when making y=y:<br /><br />-(1/4)x + (5/3) = -(1/4)x + (5/3)<br />-(1/4)x + (1/4)x = (5/3) - (5/3)<br />0=0<br /><br /><br />BUT<br /><br />even if they were not identical, you would still set them equal to each other and simplify as far as you could.<br /><br />AND<br /><br />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.<br /><br />OR<br /><br />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.<br /><br /><br />Hope that helps :)</p>