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4 x 2 y = 12 10 x + 5 y = 30 {:[4x-2y=12],[-10 x+5y=-30]:}
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We can simplify the first equation by factoring out a 2:
4 x 2 y = 12 2 ( 2 x y ) = 12 2 x y = 6 {:[4x-2y=12],[],[2(2x-y)=12],[],[2x-y=6],[]:}
Now we have the system:
2 x y = 6 10 x + 5 y = 30 {:[2x-y=6],[],[-10 x+5y=-30]:}
We can simplify the second equation by factoring out a 5:
10 x + 5 y = 30 5 ( 2 x + y ) = 30 2 x + y = 6 {:[-10 x+5y=-30],[],[5(-2x+y)=-30],[],[-2x+y=-6],[]:}
Now we have the system:
2 x y = 6 2 x + y = 6 {:[2x-y=6],[],[-2x+y=-6]:}
We can solve the second equation for y y:
2 x + y = 6 y = 2 x 6 {:[-2x+y=-6],[],[y=2x-6],[]:}
Now we can substitute this expression for y y into the first equation and solve for x x:
2 x y = 6 2 x ( 2 x 6 ) = 6 2 x 2 x + 6 = 6 6 = 6 {:[2x-y=6],[],[2x-(2x-6)=6],[],[2x-2x+6=6],[],[6=6],[]:}
This equation is true for all values of x x because we have an identity. Therefore, there are infinitely many solutions to the system. We can write the solution set as ( x , y ) = ( x , 2 x 6 ) (x,y)=(x,2x-6), where x x can be any real number.
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