How to solve this equation?!?

Your parentheses on the right side aren't necessary, so I suspect you meant to write

5 / (y + 4) = 4 + 3 / (y - 2)

But I'll solve both. First, what I think you meant.

5 / (y + 4) = 4 + 3 / (y - 2) ... multiply both sides by (y+4)(y-2) to get rid of the fractions.

5(y - 2) = 4(y + 4)(y - 2) + 3(y + 4) ... simplify

5y - 10 = 4y^2 + 8y - 32 + 3y + 12 ... simplify

5y - 10 = 4y^2 + 11y - 20 ... simplify

0 = 4y^2 + 6y - 10 ... divide both sides by 2

0 = 2y^2 + 3y - 5 ... factor

0 = (2y + 5)(y - 1) ... to get a product of zero, one of the terms on the right must be equal to zero

0 = 2y + 5 OR 0 = y - 1

2y = -5 OR y = 1

y = -5/2 OR y = 1

And that's the answer. First, note that there are two possible values for y. Either y = -5/2 or y = 1 solve the original equation. You should plug them in and check.

Second, note that when you do plug them in and check, neither of these values of y make the denominators in the original equation equal to zero, so they are both valid solutions. If plugging a value in makes a denominator zero, the expression would be undefined, and there's no way to evaluate the equality statement. But that's not the case, so both are valid.

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Now, we'll look at what you did write:

5 / (y + 4) = 4 + (3 / y - 2) ... is the same as

5 / (y + 4) = 4 + 3/y - 2 ... which simplifies to

5 / (y + 4) = 2 + 3/y ... Multiply both sides by y*(y+4) to get rid of the fractions.

5 y = 2 y(y + 4) + 3(y + 4) ... simplify

5y = 2y^2 + 8y + 3y + 12 ... simplify

0 = 2y^2 + 6y + 12 ... divide both sides by 2

0 = y^2 + 3y + 6 ... use the quadratic formula

y = [-3 ± √(3^2 - 4*1*6)] / (2*1) ... simplify

y = [-3 ± √(9 - 24)] / 2 ... simplify

y = [-3 ± √(-15)] / 2 ... simplify, recall that √-1 is the "imaginary number", represented by i

y = (-3 ± i√15) / 2

And that's the answer. First, note that there are two possible values for y. Either y = (-3 + i√15) / 2 or y = (-3 - i√15) / 2 solve the original equation.

Second, note that these values of y are not real numbers, they're complex. In fact, they're complex conjugates - that is, they both have the same real part, -3/2, and they have imaginary parts with opposite sign, i√15 / 2 and -i√15 / 2. Again, you could plug these into the original equation to check the solutions, although arithmetic with complex numbers is ... complicated. :-)

But if you did that, you'd note that neither of these values make the denominators in the original equation equal to zero, so they are both valid solutions. Again, if plugging a value in makes a denominator zero, the expression would be undefined, and there's no way to evaluate the equality statement.

I hope that helps!

5/(y+4) = 4+(3/y-2)

1. move all the variables/unknown numbers (y) onto one side of the equation

5/(y+4) (3/y-2) = 4

2. move the actual numbers to the other side (-4 +2)

5/y+ 3/y = 2

3. add up 5 and 3 to make 8

8/y = 2

4. work out what y is -8/4=2

y = 4

the answer is nineteen! specific how-to below: to make this less difficult, team the x's alongside with, of direction, their +/- indicators: 3x - 2x + 17 + 2 now that's an equation, and in case you spot no equivalent sign (=), then you rather ought to anticipate that if there become, the whole equation must be equivalent to 0: 3x - 2x + 17 + 2 = 0 now simplify-- use undemanding subtraction while coping with the variables (ignoring the x's, yet remembering to place them sooner or later result) this implies so you might respond to "3-2": 1x + 19 = 0 i've got been given the nineteen via including 17 and a couple of. you at the instant have a "1x", yet you may do away with the "a million", simply by fact writing down in basic terms "x" rather skill "1x" x + 19 = 0 now to unravel it, the variable must be on one area, and the universal cost on the different area. remember that in case you're taking a huge determination or a variable around the equivalent sign, you ought to constantly replace that's +/- sign! : x = -19 or 19 = -x now you have the answer! x is comparable to "adverse 19" or "-19" or you may study it as "adverse x is comparable to 19", that's a similar component. wish this become adequate element.