SOLUTION: solve the following system 1/4x-1/6y=1 1/2x+1/3y=2

Question 391258: solve the following system
1/4x-1/6y=1
1/2x+1/3y=2
Answer by haileytucki(390) (Show Source): You can put this solution on YOUR website!
(1)/(4x)-(1)/(6y)=1_(1)/(2x)+(1)/(3y)=2
Remove the parentheses around the 4x in the denominator.
(1)/(4x)-(1)/(6y)=1_(1)/(2x)+(1)/(3y)=2
Remove the parentheses around the 2x in the denominator.
(1)/(4x)-(1)/(6y)=1_(1)/(2x)+(1)/(3y)=2
Since -(1)/(6y) does not contain the variable to solve for, move it to the right-hand side of the equation by adding (1)/(6y) to both sides.
(1)/(4x)=(1)/(6y)+1_(1)/(2x)+(1)/(3y)=2
Simplify the right-hand side of the equation.
(1)/(4x)=(6y+1)/(6y)_(1)/(2x)+(1)/(3y)=2
Since there is one rational expression on each side of the equation, this can be solved as a ratio. For example, (A)/(B)=(C)/(D) is equivalent to A*D=B*C.
1*6y=(6y+1)*4x_(1)/(2x)+(1)/(3y)=2
Since x is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.
(6y+1)*4x=1*6y_(1)/(2x)+(1)/(3y)=2
Multiply (6y+1) by 4x to get 4x(6y+1).
4x(6y+1)=1*6y_(1)/(2x)+(1)/(3y)=2
Multiply 1 by 6y to get 6y.
4x(6y+1)=6y_(1)/(2x)+(1)/(3y)=2
Multiply 4x by each term inside the parentheses.
24xy+4x=6y_(1)/(2x)+(1)/(3y)=2
Factor out the GCF of 4x from each term in the polynomial.
4x(6y)+4x(1)=6y_(1)/(2x)+(1)/(3y)=2
Factor out the GCF of 4x from 24xy+4x.
4x(6y+1)=6y_(1)/(2x)+(1)/(3y)=2
Divide each term in the equation by (6y+1).
(4x(6y+1))/(6y+1)=(6y)/(6y+1)_(1)/(2x)+(1)/(3y)=2
Simplify the left-hand side of the equation by canceling the common factors.
4x=(6y)/(6y+1)_(1)/(2x)+(1)/(3y)=2
Divide each term in the equation by 4.
(4x)/(4)=(6y)/(6y+1)/(4)_(1)/(2x)+(1)/(3y)=2
Simplify the left-hand side of the equation by canceling the common factors.
x=(6y)/(6y+1)/(4)_(1)/(2x)+(1)/(3y)=2
Simplify the right-hand side of the equation by simplifying each term.
x=(3y)/(2(6y+1))_(1)/(2x)+(1)/(3y)=2
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is (3y)/(2(6y+1)).
x=(3y)/(2(6y+1))_(1)/(2((3y)/(2(6y+1))))+(1)/(3y)=2
Cancel the common factor of 2 from the denominator of the first expression and the numerator of the second expression.
x=(3y)/(2(6y+1))_1*(6y+1)/(3y)+(1)/(3y)=2
Multiply the rational expressions to get ((6y+1))/(3y).
x=(3y)/(2(6y+1))_(6y+1)/(3y)+(1)/(3y)=2
Divide each term in the numerator by the denominator.
x=(3y)/(2(6y+1))_(6y)/(3y)+(1)/(3y)+(1)/(3y)=2
Remove the common factors that were cancelled out.
x=(3y)/(2(6y+1))_2+(1)/(3y)+(1)/(3y)=2
Combine the numerators of all expressions that have common denominators.
x=(3y)/(2(6y+1))_2+(1+1)/(3y)=2
Add 1 to 1 to get 2.
x=(3y)/(2(6y+1))_2+(2)/(3y)=2
Find the LCD (least common denominator) of 2+(2)/(3y)+2.
x=(3y)/(2(6y+1))_Least common denominator: 3y
Multiply each term in the equation by 3y in order to remove all the denominators from the equation.
x=(3y)/(2(6y+1))_2*3y+(2)/(3y)*3y=2*3y
Simplify the left-hand side of the equation by canceling the common factors.
x=(3y)/(2(6y+1))_6y+2=2*3y
Multiply 2 by 3y to get 6y.
x=(3y)/(2(6y+1))_6y+2=6y
Since 6y contains the variable to solve for, move it to the left-hand side of the equation by subtracting 6y from both sides.
x=(3y)/(2(6y+1))_6y+2-6y=0
Since 6y and -6y are like terms, add -6y to 6y to get 0.
x=(3y)/(2(6y+1))_0+2=0
Remove the 0 from the polynomial; adding or subtracting 0 does not change the value of the expression.
x=(3y)/(2(6y+1))_2=0
Since 2 does not equal 0, there are no solutions.
No Solution
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