Question 90122

If you want to find the equation of line with a given a slope of {{{5}}} which goes through the point ({{{0}}},{{{4}}}), 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 ({{{x[1]}}},{{{y[1]}}}) is the given point


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


{{{y-4=(5)(x-0)}}} Plug in {{{m=5}}}, {{{x[1]=0}}}, and {{{y[1]=4}}} (these values are given)


{{{y-4=(5)x-(5)(0))}}} Distribute {{{5}}}


{{{y-4=(5)x+(-5)(0))}}} Multiply the negatives


{{{y-4=(5)x+0}}} Multiply {{{-5}}} and {{{0}}} to get {{{0}}}


{{{y=(5)x+0+4}}}Add {{{4}}} to both sides


{{{y=5x+4}}} Combine like terms

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Answer:



So the equation of the line with a slope of {{{5}}} which goes through the point ({{{0}}},{{{4}}}) is:


{{{y=5x+4}}} which is now in {{{y=mx+b}}} form where the slope is {{{m=5}}} and the y-intercept is {{{b=4}}}


Notice if we graph the equation {{{y=5x+4}}} and plot the point ({{{0}}},{{{4}}}),  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, -9, 9, -5, 13,
graph(500, 500, -9, 9, -5, 13,(5)x+4),
circle(0,4,0.12),
circle(0,4,0.12+0.03)
) }}} Graph of {{{y=5x+4}}} through the point ({{{0}}},{{{4}}})

and we can see that the point lies on the line. Since we know the equation has a slope of {{{5}}} and goes through the point ({{{0}}},{{{4}}}), this verifies our answer.


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If you want to find the equation of line with a given a slope of {{{3/4}}} which goes through the point ({{{0}}},{{{8}}}), 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 ({{{x[1]}}},{{{y[1]}}}) is the given point


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


{{{y-8=(3/4)(x-0)}}} Plug in {{{m=3/4}}}, {{{x[1]=0}}}, and {{{y[1]=8}}} (these values are given)


{{{y-8=(3/4)x-(3/4)(0))}}} Distribute {{{3/4}}}


{{{y-8=(3/4)x+(-3/4)(0))}}} Multiply the negatives


{{{y-8=(3/4)x+0}}} Multiply {{{-3/4}}} and {{{0}}} to get {{{0}}}


{{{y=(3/4)x+0+8}}}Add {{{8}}} to both sides


{{{y=(3/4)x+8}}} Combine like terms

------------------------------------------------------------------------------------------------------------

Answer:



So the equation of the line with a slope of {{{3/4}}} which goes through the point ({{{0}}},{{{8}}}) is:


{{{y=(3/4)x+8}}} which is now in {{{y=mx+b}}} form where the slope is {{{m=3/4}}} and the y-intercept is {{{b=8}}}


Notice if we graph the equation {{{y=(3/4)x+8}}} and plot the point ({{{0}}},{{{8}}}),  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, -9, 9, -1, 17,
graph(500, 500, -9, 9, -1, 17,(3/4)x+8),
circle(0,8,0.12),
circle(0,8,0.12+0.03)
) }}} Graph of {{{y=(3/4)x+8}}} through the point ({{{0}}},{{{8}}})

and we can see that the point lies on the line. Since we know the equation has a slope of {{{3/4}}} and goes through the point ({{{0}}},{{{8}}}), this verifies our answer.