SOLUTION: Hello out there,I'm writing for some help with some slope-intercept problems,the first one is,Are the following two graphs in a family?If so/explain why,y=3x-2 and y=-2x+1.The next

Algebra ->  Linear-equations -> SOLUTION: Hello out there,I'm writing for some help with some slope-intercept problems,the first one is,Are the following two graphs in a family?If so/explain why,y=3x-2 and y=-2x+1.The next      Log On


   

Question 83198: Hello out there,I'm writing for some help with some slope-intercept problems,the first one is,Are the following two graphs in a family?If so/explain why,y=3x-2 and y=-2x+1.The next one is,Are the following two graphs in a family?If so explain why,y=-2x+3 and y=2x+3.The next one is,Change y=x-3 so that the graph of the new equation has a steeper,negative slope the same y-intercept.And the last one is,Write the point-slope form of the line that passes through (3,-3)and(5,1),thanks for whoever can help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If you graph y=3x-2+ you get
+graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+3x-2%29+

If you graph y=-2x%2B1+ you get
+graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C-2x%2B1%29+

Since neither of these graphs share the same slope or y-intercept, they are not in a family (ie they have nothing in common).
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If you graph y=-2x%2B3+ you get
+graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C-2x%2B3%29+


and if you graph y=2x%2B3+ you get
+graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C2x%2B3%29+

You will notice that these two graphs have the same y-intercept, so they belong in a family (since they share the same y-intercept).


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If you have y=x-3 and you want a steeper, negative slope, simply make the slope a large negative value (for the general slope-intercept equation y=mx%2Bb, m is the slope) . So I could choose -7, and that would make a steep, negative slope. So if I graphed y=-7x-3 I would get

+graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+-7x-3%29+

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Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (3,-3) and (5,1)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (3,-3) and (x%5B2%5D,y%5B2%5D) is the second point (5,1))


m=%281--3%29%2F%285-3%29 Plug in y%5B2%5D=1,y%5B1%5D=-3,x%5B2%5D=5,x%5B1%5D=3 (these are the coordinates of given points)


m=+4%2F2 Subtract the terms in the numerator 1--3 to get 4. Subtract the terms in the denominator 5-3 to get 2




m=2 Reduce



So the slope is

m=2





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Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and (x%5B1%5D,y%5B1%5D) is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


y--3=%282%29%28x-3%29 Plug in m=2, x%5B1%5D=3, and y%5B1%5D=-3 (these values are given)



y%2B3=%282%29%28x-3%29 Rewrite y--3 as y%2B3



y%2B3=2x%2B%282%29%28-3%29 Distribute 2


y%2B3=2x-6 Multiply 2 and -3 to get -6%2F1. Now reduce -6%2F1 to get -6

y=2x-6-3 Subtract 3 from both sides to isolate y


y=2x-9 Combine like terms -6 and -3 to get -9

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Answer:



So the equation of the line which goes through the points (3,-3) and (5,1) is:y=2x-9


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=2 and the y-intercept is b=-9


Notice if we graph the equation y=2x-9 and plot the points (3,-3) and (5,1), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=2x-9 through the points (3,-3) and (5,1)


Notice how the two points lie on the line. This graphically verifies our answer.