Question 172154
# 1


If you want to find the equation of line with a given a slope of {{{2/3}}} which goes through the point ({{{4}}},{{{1}}}), you can simply use the point-slope formula to find the equation:



---Point-Slope Formula---
{{{y-y[1]=m(x-x[1])}}} where {{{m}}} is the slope, and *[Tex \Large \left(x_{1},y_{1}\right)] is the given point


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


{{{y-1=(2/3)(x-4)}}} Plug in {{{m=2/3}}}, {{{x[1]=4}}}, and {{{y[1]=1}}} (these values are given)



{{{y-1=(2/3)x+(2/3)(-4)}}} Distribute {{{2/3}}}


{{{y-1=(2/3)x-8/3}}} Multiply {{{2/3}}} and {{{-4}}} to get {{{-8/3}}}


{{{y=(2/3)x-8/3+1}}} Add 1 to  both sides to isolate y


{{{y=(2/3)x-5/3}}} Combine like terms {{{-8/3}}} and {{{1}}} to get {{{-5/3}}} (note: if you need help with combining fractions, check out this <a href=http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver>solver</a>)



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Answer:



So the equation of the line with a slope of {{{2/3}}} which goes through the point ({{{4}}},{{{1}}}) is:


{{{y=(2/3)x-5/3}}} which is now in {{{y=mx+b}}} form where the slope is {{{m=2/3}}} and the y-intercept is {{{b=-5/3}}}


Notice if we graph the equation {{{y=(2/3)x-5/3}}} and plot the point ({{{4}}},{{{1}}}),  we get (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)


{{{drawing(500, 500, -5, 13, -8, 10,
graph(500, 500, -5, 13, -8, 10,(2/3)x+-5/3),
circle(4,1,0.12),
circle(4,1,0.12+0.03)
) }}} Graph of {{{y=(2/3)x-5/3}}} 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 {{{2/3}}} and goes through the point ({{{4}}},{{{1}}}), this verifies our answer.




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# 2




First let's find the slope of the line through the points *[Tex \LARGE \left(7,3\right)] and *[Tex \LARGE \left(-2,9\right)]



{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(9-3)/(-2-7)}}} Plug in {{{y[2]=9}}}, {{{y[1]=3}}}, {{{x[2]=-2}}}, and {{{x[1]=7}}}



{{{m=(6)/(-2-7)}}} Subtract {{{3}}} from {{{9}}} to get {{{6}}}



{{{m=(6)/(-9)}}} Subtract {{{7}}} from {{{-2}}} to get {{{-9}}}



{{{m=-2/3}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(7,3\right)] and *[Tex \LARGE \left(-2,9\right)] is {{{m=-2/3}}}



Now let's use the point slope formula:



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-3=(-2/3)(x-7)}}} Plug in {{{m=-2/3}}}, {{{x[1]=7}}}, and {{{y[1]=3}}}



{{{y-3=(-2/3)x+(-2/3)(-7)}}} Distribute



{{{y-3=(-2/3)x+14/3}}} Multiply



{{{y=(-2/3)x+14/3+3}}} Add 3 to both sides. 



{{{y=(-2/3)x+23/3}}} Combine like terms. note: If you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>.



{{{y=(-2/3)x+23/3}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(7,3\right)] and *[Tex \LARGE \left(-2,9\right)] is {{{y=(-2/3)x+23/3}}}


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# 3


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





First let's find the slope of the line through the points *[Tex \LARGE \left(17,5\right)] and *[Tex \LARGE \left(7,0\right)]



{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(0-5)/(7-17)}}} Plug in {{{y[2]=0}}}, {{{y[1]=5}}}, {{{x[2]=7}}}, and {{{x[1]=17}}}



{{{m=(-5)/(7-17)}}} Subtract {{{5}}} from {{{0}}} to get {{{-5}}}



{{{m=(-5)/(-10)}}} Subtract {{{17}}} from {{{7}}} to get {{{-10}}}



{{{m=1/2}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(17,5\right)] and *[Tex \LARGE \left(7,0\right)] is {{{m=1/2}}}



Now let's use the point slope formula:



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-5=(1/2)(x-17)}}} Plug in {{{m=1/2}}}, {{{x[1]=17}}}, and {{{y[1]=5}}}



{{{y-5=(1/2)x+(1/2)(-17)}}} Distribute



{{{y-5=(1/2)x-17/2}}} Multiply



{{{y=(1/2)x-17/2+5}}} Add 5 to both sides. 



{{{y=(1/2)x-7/2}}} Combine like terms. note: If you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>.



{{{y=(1/2)x-7/2}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(17,5\right)] and *[Tex \LARGE \left(7,0\right)] is {{{y=(1/2)x-7/2}}}