Question 228442
Suppose the line L has a slope of (-3/5) and passes through the points (11, t) and (2t, -8 ). Find the equation of L and its x and y intercepts.


Step 1.  The slope-intercept form is given as y=mx+b where m is the slope and b is the y-intercept b at x=0 or point (0,b).  Here, the slope m=-3/5.


Step 2.  The slope m is given as


{{{m=(y2-y1)/(x2-x1)}}}


Step 3.  Let (x1,y1)=(11,t) or x1=11 and y1=t and (x2,y2)=(2t,-8) or x2=2t and y2=-8.


Step 4.  Now we're given {{{m=-3/5}}}.  Substituting above values and variables in the slope equation m yields the following steps:


{{{m=(y2-y1)/(x2-x1)}}}


{{{-3/5=(-8-t)/(2t-11)}}}


Step 5.  Multiply 5(2t-11) to both sides to get rid of denominators on both sides of equation.


{{{5(2t-11)*(-3/5)=5(2t-11)(-8-t)/(2t-11)}}} 


{{{-3(2t-11)=5(-8-t)}}} 


{{{-6t+33=-40-5t}}}


Add 6t+40 to both sides of the equation


{{{-6t+33+6t+40=-40-5t+6t+40}}}


{{{73=t}}}


Step 6.  Therefore the points are (x1,y1)=(11,73) and (x2,y2)=(2*73,-8) or (x2,y2)=(146,-8)


Let's double check the slope m=-3/5 with these two points (11,73) and (146,-8).


{{{m=(-8-73)/(146-11)=-81/135=-3/5}}} so it works.


Step 7.  Now to get our linear equation,  we choose one of the points say (11,73) and the slope {{{m=-3/5}}}.  We have (x1,y1)=(11,73) or x1=11 and y1=73.  Let other point be (x2,y2)=(x,y) or x2=x and y2=y.


Step 8.  Now we're given {{{m=-3/5}}}.  Substituting above values and variables in the slope equation m yields the following steps to get a linear equation:


{{{m=(y2-y1)/(x2-x1)}}}


{{{-3/5=(y-73)/(x-11)}}}


Multiply by (x-11) to both sides of the equation


{{{-3(x-11)/5=y-73}}}


{{{-3x/5+33/5=y-73}}}


Add 73 to both sides of the equation


{{{-3x/5+33/5+73=y-73+73}}}


{{{-3x/5+33/5+365/5=y}}}


{{{-3x/5+398/5=y}}}


Step 9.  Let's see if the other point (146,-8) satisfies {{{y=-3x/5+398/5}}} as a check


{{{-8=-3*146/5+398/5}}}


{{{-8=-438/5+398/5=-40/5}}}


{{{-8=-8}}}  which is a true statement


Step 10.  The equation is {{{y=-3x/5+398/5}}}.  


Step 11.  The x-intercept is when y=0 or


{{{0=-3x/5+398/5}}} after add 3x/5 and multiplying by 5 to both sides of the equation we have


{{{3x=398}}}


{{{x=398/3}}} ANSWER to finding the x-intercept.


Step 12.  The y-intercept is when x=0 or


{{{y=-3*0/5+398/5}}} after add 3x/5 and multiplying by 5 to both sides of the equation we have


{{{y=398/5}}}


{{{y=398/5}}} ANSWER to finding the y-intercept.


Step 13.  ANSWER:  The equation is {{{y=-3x/5+398/5}}}.  The x-intercept is {{{a=398/3}}} or at point ({{{398/3}}}, {{{0}}}).  The y-intercept is {{{b=398/5}}} or at point ({{{0}}},{{{398/5}}}).


I hope the above steps were helpful.


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Good luck in your studies!


Respectfully,
Dr J