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Question 1203089: x*y=2x-y find the value of y if y*(3*y)=6
Found 4 solutions by Alan3354, Theo, math_tutor2020, ikleyn:
Answer by Alan3354(69443)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! your two equations are:
x*y=2x-y
y*(3*y)=6
simplify to get:
x*y = 2x - y
3y^2 = 6
in the first equation:
add y to both sides of the equation to get x*y + y = 2x
factor out the y to get:
y * (x + 1) = 2x
solve for y to get:
y = 2x / (x + 1)
in the second equation solve for y^2 to get:
y^2 = 2
since y = 2x / (x + 1), then:
y^2 = 4x^2 / (x^2 + 2x + 1)
your equation of y^2 = 2 becomes:
4x^2 / (x^2 + 2x + 1) = 2
multiply both sides of this equation by (x^2 + 2x + 1) to get:
4x^2 = 2 * (x^2 + 2x + 1)
simplify to ge:
4x^2 = 2x^2 + 4x + 2
subtract the right hand side of the equation from both sides of the equation to get:
4x^2 - 2x^2 - 4x - 2 = 0
combine like terms to get:
2x^2 - 4x - 2 = 0
factor out a 2 to get:
x^2 - 2x - 1 = 0
factor this quadratic equation to get:
x = 2.4142135623731 or x = -0.4142135623731
to confirm, replace x with either of these values in the original equation to solve the original problem.
the original problem gives you the original equation of x*y=2x-y.
it then asks you to find the value of y if y * (3 * y) = 6
i think it might be asking you to find the value of x is y * (3 * y) = 6
i solved for x to get the values of x above.
i also solved for y in the second equation to get y^2 = 2
thqat would make y equal to plus or minus sqrt(2).
using the values of x above, i replaced x in the original equation of x*y = (2x -y,
solving for y in that equation, i got y = 2x / (x + 1).
using the value of x i derived above, i got y = plus or minus sqrt(2).
i graphed both equation and got what you see below:
your solution is that y = sqrt(2) when x = 2.4142135623731 and y = -sqrt(2) when x = -0.4142135623731.
alternatively, your solution is that x = 2.4142135623731 when y = sqrt(2) and x = -0.4142135623731 when y = -sqrt(2).
i used a quadratic solver to find the values of x.
here are the results from that quadratic equation solver.
not that y = -1.414 and y 1.414 that you see on the graph is a rounded version of y = -sqrt(2) and y = sqrt(2).
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
In many other contexts, the asterisk or star symbol * indicates multiplication.
Example: 2*3 = 6
However, this problem has the star operator redefined as:
x*y = 2x-y
This is a user-defined operator.
It is not to be confused with the multiplication symbol.
So be careful not to assume that y*(3*y) turns into 3y^2
Let's replace x with 3
x*y = 2x-y
3*y = 2(3)-y
3*y = 6-y
This means any time we see 3*y, we can replace it with 6-y.
x*y = 2x-y
x*(3*y) = 2x-(3*y)
x*(3*y) = 2x-(6-y)
x*(3*y) = 2x-6+y
x*(3*y) = 2x+y-6
Next we'll replace x with y so we end up with y*(3*y)
x*(3*y) = 2x+y-6
y*(3*y) = 2y+y-6
y*(3*y) = 3y-6
We're told that y*(3*y) = 6
We'll use the previous equation and this given fact to solve for y.
y*(3*y) = 6
3y-6 = 6
3y = 6+6
3y = 12
y = 12/3
y = 4
Answer: y = 4
Answer by ikleyn(52781)  (Show Source):
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