Question 145021
# 1






Start with the given system of equations:


{{{system(5x+6y=14,x-3y=7)}}}





{{{1(5x+6y)=1(14)}}}  Multiply the top equation (both sides) by {{{1}}}
{{{-5(x-3y)=-5(7)}}}  Multiply the bottom equation (both sides) by {{{-5}}}





Distribute and multiply


{{{5x+6y=14}}}
{{{-5x+15y=-35}}}



Now add the equations together. In order to add 2 equations, group like terms and combine them


{{{(5x-5x)+(6y+15y)=14-35}}}


Combine like terms and simplify




{{{cross(5x-5x)+21y=-21}}} Notice how the x terms cancel out





{{{21y=-21}}} Simplify





{{{y=-21/21}}} Divide both sides by {{{21}}} to isolate y





{{{y=-1}}} Reduce




Now plug this answer into the top equation {{{5x+6y=14}}} to solve for x


{{{5x+6y=14}}} Start with the first equation




{{{5x+6(-1)=14}}} Plug in {{{y=-1}}}





{{{5x-6=14}}} Multiply




{{{5x=14+6}}}Add 6 to both sides



{{{5x=20}}} Combine like terms on the right side



{{{x=(20)/(5)}}} Divide both sides by 5 to isolate x




{{{x=4}}} Divide





So our answer is

{{{x=4}}} and {{{y=-1}}}




which also looks like *[Tex \LARGE \left(4,-1\right)]





Now let's graph the two equations (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>)



From the graph, we can see that the two equations intersect at *[Tex \LARGE \left(4,-1\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (14-5*x)/(6), (7-1*x)/(-3) ),
  blue(circle(4,-1,0.1)),
  blue(circle(4,-1,0.12)),
  blue(circle(4,-1,0.15))
)
}}} graph of {{{5x+6y=14}}} (red) and {{{x-3y=7}}} (green)  and the intersection of the lines (blue circle).








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# 2





Let's look at the first equation  {{{2x/3+3y/4=11/12}}}



{{{12(2x/cross(3)+3y/cross(4))=12(11/cross(12))}}} Multiply both sides of the first equation by the LCD 12



{{{8x+9y=11}}} Distribute



---------



Let's look at the second equation  {{{x/3+7y/18=1/2}}}


{{{18(x/cross(3)+7y/cross(18))=18(1/cross(2))}}} Multiply both sides of the second equation by the LCD 18



{{{6x+7y=9}}} Distribute



---------




So our new system of equations is:



{{{system(8x+9y=11,6x+7y=9)}}}





{{{3(8x+9y)=3(11)}}}  Multiply the top equation (both sides) by {{{3}}}
{{{-4(6x+7y)=-4(9)}}}  Multiply the bottom equation (both sides) by {{{-4}}}





Distribute and multiply


{{{24x+27y=33}}}
{{{-24x-28y=-36}}}



Now add the equations together. In order to add 2 equations, group like terms and combine them


{{{(24x-24x)+(27y-28y)=33-36}}}


Combine like terms and simplify




{{{cross(24x-24x)-y=-3}}} Notice how the x terms cancel out





{{{-y=-3}}} Simplify





{{{y=-3/-1}}} Divide both sides by {{{-1}}} to isolate y





{{{y=3}}} Reduce




Now plug this answer into the top equation {{{8x+9y=11}}} to solve for x


{{{8x+9y=11}}} Start with the first equation




{{{8x+9(3)=11}}} Plug in {{{y=3}}}





{{{8x+27=11}}} Multiply




{{{8x=11-27}}}Subtract 27 from both sides



{{{8x=-16}}} Combine like terms on the right side



{{{x=(-16)/(8)}}} Divide both sides by 8 to isolate x




{{{x=-2}}} Divide





So our answer is

{{{x=-2}}} and {{{y=3}}}




which also looks like *[Tex \LARGE \left(-2,3\right)]