How do you solve #\frac { 6y } { 2y + 6} + \frac { y + 18} { 3y + 9} = \frac { 7y + 17} { y + 3}#?
No real solution. Read the explanation carefully.
We can rewrite it as:
Cross multiply;
Taking
BUT there is a catch.
We might think that
We find that terms in the denominator become zero.
That means the expressions
Where did we go wrong?
Remember the step where we cross multiplied
Just to be sure, I checked on wolfram.
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To solve the equation (\frac{6y}{2y+6} + \frac{y+18}{3y+9} = \frac{7y+17}{y+3}), we can follow these steps:
 Simplify the fractions on both sides of the equation.
 Find a common denominator for the fractions.
 Combine the fractions on both sides of the equation.
 Solve for the variable, y.
Let's go through each step in detail:

Simplify the fractions:
 The fraction (\frac{6y}{2y+6}) cannot be simplified further.
 The fraction (\frac{y+18}{3y+9}) can be simplified by factoring out a common factor of 3 from the numerator and denominator, resulting in (\frac{y+18}{3(y+3)}).
 The fraction (\frac{7y+17}{y+3}) cannot be simplified further.

Find a common denominator:
 The denominators in the equation are (2y+6), (3(y+3)), and (y+3).
 The common denominator for these fractions is (3(y+3)).

Combine the fractions:
 Rewrite the equation with the common denominator: (\frac{6y}{2y+6} + \frac{y+18}{3(y+3)} = \frac{7y+17}{y+3}).
 Multiply each fraction by the necessary factors to obtain the common denominator: (\frac{6y}{2y+6} \cdot \frac{3(y+3)}{3(y+3)} + \frac{y+18}{3(y+3)} \cdot \frac{2y+6}{2y+6} = \frac{7y+17}{y+3}).
 Simplify the numerators: (\frac{18y}{6y+18} + \frac{(y+18)(2y+6)}{3(y+3)(2y+6)} = \frac{7y+17}{y+3}).

Solve for the variable, y:
 Multiply both sides of the equation by the common denominator to eliminate the fractions: (18y + (y+18)(2y+6) = (7y+17)(3(y+3)(2y+6))).
 Expand and simplify both sides of the equation: (18y + 2y^2 + 42y + 126 = 42y^3 + 189y^2 + 441y + 306).
 Rearrange the equation to form a cubic equation: (42y^3 + 189y^2 + 441y + 306  18y  2y^2  42y  126 = 0).
 Combine like terms: (42y^3 + 187y^2 + 381y + 180 = 0).
 Solve the cubic equation using numerical methods or factoring techniques.
Please note that solving the cubic equation may require advanced mathematical techniques and may not have a simple algebraic solution.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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