SOLUTION: Hello; I wondered if you could explain how to answer this question: Solve each system by graphing. y=-2/3x and 2x+3y=5

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Question 321296: Hello; I wondered if you could explain how to answer this question:
Solve each system by graphing. y=-2/3x and 2x+3y=5


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First let's graph y=%28-2%2F3%29x


Looking at y=-%282%2F3%29x 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=0 note: y=-%282%2F3%29x really looks like y=-%282%2F3%29x%2B0


Since b=0 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 down 2 and over 3



So starting at , go down 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

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



Now let's graph 2x%2B3y=5. To do this, we first must solve for 'y'.



2x%2B3y=5 Start with the second equation.


3y=5-2x Subtract 2x from both sides.


3y=-2x%2B5 Rearrange the terms.


y=%28-2x%2B5%29%2F%283%29 Divide both sides by 3 to isolate y.


y=%28%28-2%29%2F%283%29%29x%2B%285%29%2F%283%29 Break up the fraction.


y=-%282%2F3%29x%2B5%2F3 Reduce.


Looking at y=-%282%2F3%29x%2B5%2F3 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=5%2F3


Since b=5%2F3 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 down 2 and over 3



So starting at , go down 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%2B5%2F3

So this is the graph of y=-%282%2F3%29x%2B5%2F3 through the points and


Since 2x%2B3y=5 is equivalent to y=-%282%2F3%29x%2B5%2F3, this means that the graph shown above is also the graph of 2x%2B3y=5

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Now graph the two equations on the same coordinate plane:




Graph of y=-%282%2F3%29x (red) and 2x%2B3y=5 (green)


From the graph, the two lines are parallel and will NEVER intersect. So there are NO solutions. This means that the system is inconsistent.