SOLUTION: solve each system of equations and graph then identify if the system is inconsistent or dependent and show why y=1/2x+2 y=2/3x-1

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Question 243642: solve each system of equations and graph then identify if the system is inconsistent or dependent and show why
y=1/2x+2
y=2/3x-1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First let's graph y=%281%2F2%29x%2B2


Looking at y=%281%2F2%29x%2B2 we can see that the equation is in slope-intercept form y=mx%2Bb where the slope is m=1%2F2 and the y-intercept is b=2


Since b=2 this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means
slope=rise%2Frun

Also, because the slope is 1%2F2, this means:

rise%2Frun=1%2F2


which shows us that the rise is 1 and the run is 2. This means that to go from point to point, we can go up 1 and over 2



So starting at , go up 1 unit


and to the right 2 units to get to the next point



Now draw a line through these points to graph y=%281%2F2%29x%2B2

So this is the graph of y=%281%2F2%29x%2B2 through the points and


Now let's graph y=%282%2F3%29x-1



Looking at y=%282%2F3%29x-1 we can see that the equation is in slope-intercept form y=mx%2Bb where the slope is m=2%2F3 and the y-intercept is b=-1


Since b=-1 this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means
slope=rise%2Frun

Also, because the slope is 2%2F3, this means:

rise%2Frun=2%2F3


which shows us that the rise is 2 and the run is 3. This means that to go from point to point, we can go up 2 and over 3



So starting at , go up 2 units


and to the right 3 units to get to the next point



Now draw a line through these points to graph y=%282%2F3%29x-1

So this is the graph of y=%282%2F3%29x-1 through the points and


Now let's plot the two equations together:


Graph of y=%281%2F2%29x%2B2 (red) and y=%282%2F3%29x-1 (green)


From the graph, we see that the two lines intersect at the point (18,11). So the solution is (18,11) which means that x=18 and y=11.


So the system is consistent and independent.