SOLUTION: 1. write an equation of the line that passes through (4,1) and has a slope of 2/3? 2.write an equation of the line that passes through (7,3) and (-2,9). 3. write an equation

Algebra ->  Linear-equations -> SOLUTION: 1. write an equation of the line that passes through (4,1) and has a slope of 2/3? 2.write an equation of the line that passes through (7,3) and (-2,9). 3. write an equation       Log On


   

Question 172154This question is from textbook
: 1. write an equation of the line that passes through (4,1) and has a slope of 2/3?
2.write an equation of the line that passes through (7,3) and (-2,9).
3. write an equation of the line that passes through (17,5) and intercepts the x axis at 7.
This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

If you want to find the equation of line with a given a slope of 2%2F3 which goes through the point (4,1), you can simply use the point-slope formula to find the equation:


---Point-Slope Formula---
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and is the given point

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

y-1=%282%2F3%29%28x-4%29 Plug in m=2%2F3, x%5B1%5D=4, and y%5B1%5D=1 (these values are given)


y-1=%282%2F3%29x%2B%282%2F3%29%28-4%29 Distribute 2%2F3

y-1=%282%2F3%29x-8%2F3 Multiply 2%2F3 and -4 to get -8%2F3

y=%282%2F3%29x-8%2F3%2B1 Add 1 to both sides to isolate y

y=%282%2F3%29x-5%2F3 Combine like terms -8%2F3 and 1 to get -5%2F3 (note: if you need help with combining fractions, check out this solver)


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Answer:


So the equation of the line with a slope of 2%2F3 which goes through the point (4,1) is:

y=%282%2F3%29x-5%2F3 which is now in y=mx%2Bb form where the slope is m=2%2F3 and the y-intercept is b=-5%2F3

Notice if we graph the equation y=%282%2F3%29x-5%2F3 and plot the point (4,1), we get (note: if you need help with graphing, check out this solver)

Graph of y=%282%2F3%29x-5%2F3 through the point (4,1)
and we can see that the point lies on the line. Since we know the equation has a slope of 2%2F3 and goes through the point (4,1), this verifies our answer.






# 2



First let's find the slope of the line through the points and


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula.


m=%289-3%29%2F%28-2-7%29 Plug in y%5B2%5D=9, y%5B1%5D=3, x%5B2%5D=-2, and x%5B1%5D=7


m=%286%29%2F%28-2-7%29 Subtract 3 from 9 to get 6


m=%286%29%2F%28-9%29 Subtract 7 from -2 to get -9


m=-2%2F3 Reduce


So the slope of the line that goes through the points and is m=-2%2F3


Now let's use the point slope formula:


y-y%5B1%5D=m%28x-x%5B1%5D%29 Start with the point slope formula


y-3=%28-2%2F3%29%28x-7%29 Plug in m=-2%2F3, x%5B1%5D=7, and y%5B1%5D=3


y-3=%28-2%2F3%29x%2B%28-2%2F3%29%28-7%29 Distribute


y-3=%28-2%2F3%29x%2B14%2F3 Multiply


y=%28-2%2F3%29x%2B14%2F3%2B3 Add 3 to both sides.


y=%28-2%2F3%29x%2B23%2F3 Combine like terms. note: If you need help with fractions, check out this solver.


y=%28-2%2F3%29x%2B23%2F3 Simplify


So the equation that goes through the points and is y=%28-2%2F3%29x%2B23%2F3




# 3

Note: the x intercept 7 is simply the point (7,0)




First let's find the slope of the line through the points and


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula.


m=%280-5%29%2F%287-17%29 Plug in y%5B2%5D=0, y%5B1%5D=5, x%5B2%5D=7, and x%5B1%5D=17


m=%28-5%29%2F%287-17%29 Subtract 5 from 0 to get -5


m=%28-5%29%2F%28-10%29 Subtract 17 from 7 to get -10


m=1%2F2 Reduce


So the slope of the line that goes through the points and is m=1%2F2


Now let's use the point slope formula:


y-y%5B1%5D=m%28x-x%5B1%5D%29 Start with the point slope formula


y-5=%281%2F2%29%28x-17%29 Plug in m=1%2F2, x%5B1%5D=17, and y%5B1%5D=5


y-5=%281%2F2%29x%2B%281%2F2%29%28-17%29 Distribute


y-5=%281%2F2%29x-17%2F2 Multiply


y=%281%2F2%29x-17%2F2%2B5 Add 5 to both sides.


y=%281%2F2%29x-7%2F2 Combine like terms. note: If you need help with fractions, check out this solver.


y=%281%2F2%29x-7%2F2 Simplify


So the equation that goes through the points and is y=%281%2F2%29x-7%2F2