Question 152235

{{{(3/4)x-(2/5)y=2}}} Start with the first equation.



{{{20((3/cross(4))x-(2/cross(5))y)=20(2)}}} Multiply both sides by the LCD {{{20}}} to clear any fractions.



{{{15x-8y=40}}} Distribute and multiply.



{{{(1/2)x-(3/5)y=-2}}} Move onto the second equation.



{{{10((1/cross(2))x-(3/cross(5))y)=10(-2)}}} Multiply both sides by the LCD {{{10}}} to clear any fractions.



{{{5x-6y=-20}}} Distribute and multiply.



So we have the system of equations:



{{{system(15x-8y=40,5x-6y=-20)}}}




{{{-3(5x-6y)=-3(-20)}}} Multiply the both sides of the second equation by -3.



{{{-15x+18y=60}}} Distribute and multiply.



So we have the new system of equations:

{{{system(15x-8y=40,-15x+18y=60)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(15x-8y)+(-15x+18y)=(40)+(60)}}}



{{{(15x-15x)+(-8y+18y)=40+60}}} Group like terms.



{{{0x+10y=100}}} Combine like terms. Notice how the x terms cancel out.



{{{10y=100}}} Simplify.



{{{y=(100)/(10)}}} Divide both sides by {{{10}}} to isolate {{{y}}}.



{{{y=10}}} Reduce.



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{{{15x-8y=40}}} Now go back to the first equation.



{{{15x-8(10)=40}}} Plug in {{{y=10}}}.



{{{15x-80=40}}} Multiply.



{{{15x=40+80}}} Add {{{80}}} to both sides.



{{{15x=120}}} Combine like terms on the right side.



{{{x=(120)/(15)}}} Divide both sides by {{{15}}} to isolate {{{x}}}.



{{{x=8}}} Reduce.



So our answer is {{{x=8}}} and {{{y=10}}}.



Which form the ordered pair *[Tex \LARGE \left(8,10\right)].



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(8,10\right)]. So this visually verifies our answer.



{{{drawing(500,500,-2,18,-2,20,
grid(1),
graph(500,500,-2,18,-2,20,(40-15x)/(-8),(-20-5x)/(-6)),
circle(8,10,0.05),
circle(8,10,0.08),
circle(8,10,0.10)
)}}} Graph of {{{(3/4)x-(2/5)y=2}}} (red) and {{{(1/2)x-(3/5)y=-2}}} (green)