SOLUTION: Plot the graph of the equations 4x - 6y = 2 and 2x - 3y = 1 and interpret the result. I am having such a difficult time solving this equation. Can someone please help me underst

Algebra ->  Linear-equations -> SOLUTION: Plot the graph of the equations 4x - 6y = 2 and 2x - 3y = 1 and interpret the result. I am having such a difficult time solving this equation. Can someone please help me underst      Log On


   

Question 170422: Plot the graph of the equations 4x - 6y = 2 and 2x - 3y = 1 and interpret the result.
I am having such a difficult time solving this equation. Can someone please help me understand how to do this problem, any help will be greatly appreciated.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Graphing the first equation




In order to graph 4x-6y=2, we need to solve for "y" first.


4x-6y=2 Start with the first equation.


-6y=2-4x Subtract 4x from both sides.


-6y=-4x%2B2 Rearrange the terms.


y=%28-4x%2B2%29%2F%28-6%29 Divide both sides by -6 to isolate y.


y=%28%28-4%29%2F%28-6%29%29x%2B%282%29%2F%28-6%29 Break up the fraction.


y=%282%2F3%29x-1%2F3 Reduce.




In order to graph this equation, we only need two points to create a straight line




--------------------------------Let's find the first point--------------------------------

y=%282%2F3%29x-1%2F3 Start with the given equation




y=%282%2F3%29%285%29-1%2F3 Plug in x=5




y=10%2F3-1%2F3 Multiply 2%2F3 and 5 to get 10%2F3




y=9%2F3 Subtract



So when x=5, we have the value y=3. This means we have the first point




--------------------------------Let's find the second point--------------------------------

y=%282%2F3%29x-1%2F3 Start with the given equation




y=%282%2F3%29%28-1%29-1%2F3 Plug in x=-1




y=-2%2F3-1%2F3 Multiply 2%2F3 and -1 to get -2%2F3




y=-1 Subtract



So when x=-1, we have the value y=-1. This means we have the second point




------------------------------------------------------------------------------------------------


So we have the two points: and


Now plot these two points on a coordinate system





Now draw a straight line through the two points. This line is the graph of y=%282%2F3%29x-1%2F3

Graph of y=%282%2F3%29x-1%2F3 through the two points and





Graphing the second equation




2x-3y=1 Start with the second equation.


-3y=1-2x Subtract 2x from both sides.


-3y=-2x%2B1 Rearrange the terms.


y=%28-2x%2B1%29%2F%28-3%29 Divide both sides by -3 to isolate y.


y=%28%28-2%29%2F%28-3%29%29x%2B%281%29%2F%28-3%29 Break up the fraction.


y=%282%2F3%29x-1%2F3 Reduce.


Take note that this equation is EXACTLY identical to the equation we just plotted. So the graph of y=%282%2F3%29x-1%2F3 (and 2x-3y=1) is






Now graphing the two equations together gives you




Note: there are really two equations here. One is just right on top of the other (which hides it)


Since one equation is right on top of the other, this tells us that there are an infinite number of solutions (since there are an infinite number of intersections).


So the system is dependent.