SOLUTION: HI!! please help me with this Find the equation of a line with the slope of 2/3 that passes through the points (4,-1) ps.the answer is 2/3x-11/3 but i dont know how to get

Algebra ->  Linear-equations -> SOLUTION: HI!! please help me with this Find the equation of a line with the slope of 2/3 that passes through the points (4,-1) ps.the answer is 2/3x-11/3 but i dont know how to get       Log On


   

Question 109124: HI!!
please help me with this
Find the equation of a line with the slope of 2/3 that passes through the points (4,-1)
ps.the answer is 2/3x-11/3
but i dont know how to get it

Found 2 solutions by MathLover1, jim_thompson5910:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Solved by pluggable solver: FIND a line by slope and one point

What we know about the line whose equation we are trying to find out:

  • it goes through point (4, -1)

  • it has a slope of 0.666666666666667



First, let's draw a diagram of the coordinate system with point (4, -1) plotted with a little blue dot:



Write this down: the formula for the equation, given point x%5B1%5D%2C+y%5B1%5D and intercept a, is

y=ax+%2B+%28y%5B1%5D-a%2Ax%5B1%5D%29 (see a paragraph below explaining why this formula is correct)

Given that a=0.666666666666667, and system%28+x%5B1%5D+=+4%2C+y%5B1%5D+=+-1+%29+, we have the equation of the line:

y=0.666666666666667%2Ax+%2B+-3.66666666666667

Explanation: Why did we use formula y=ax+%2B+%28y%5B1%5D+-+a%2Ax%5B1%5D%29 ? Explanation goes here. We are trying to find equation y=ax+b. The value of slope (a) is already given to us. We need to find b. If a point (x%5B1%5D, y%5B1%5D) lies on the line, it means that it satisfies the equation of the line. So, our equation holds for (x%5B1%5D, y%5B1%5D): y%5B1%5D+=+a%2Ax%5B1%5D%2Bb Here, we know a, x%5B1%5D, and y%5B1%5D, and do not know b. It is easy to find out: b=y%5B1%5D-a%2Ax%5B1%5D. So, then, the equation of the line is: +y=ax%2B%28y%5B1%5D-a%2Ax%5B1%5D%29+.

Here's the graph:

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

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%2B1=%282%2F3%29%28x-4%29 Rewrite y--1 as y%2B1


y%2B1=%282%2F3%29x%2B%282%2F3%29%28-4%29 Distribute 2%2F3

y%2B1=%282%2F3%29x-8%2F3 Multiply 2%2F3 and -4 to get -8%2F3

y=%282%2F3%29x-8%2F3-1 Subtract 1 from both sides to isolate y

y=%282%2F3%29x-11%2F3 Combine like terms -8%2F3 and -1 to get -11%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-11%2F3 which is now in y=mx%2Bb form where the slope is m=2%2F3 and the y-intercept is b=-11%2F3

Notice if we graph the equation y=%282%2F3%29x-11%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-11%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.