To solve an equation for a variable, you isolate that variable on one side of the equation and get everything else on the opposite side of the variable.
You isolate the variable by undoing the operations done to the variable.
For, example: if your variable x is being multiplied by 3, which would look like 3x, then you would undo that operation by using the inverse of multiplication. Which is division.
Also, remember this: if you do one operation on one side of the equation, you must also do the same operation to the opposite side.
[tex] \frac{3(y + 3)}{y + 1} + 2 = \frac{3y + 1}{y + 1} [/tex]
The variable y is on both sides of the equation. We need to get them on one side the equation. But first, get rid of the denominators.
We can get rid of the denominators by multiplying both sides by y + 1.
[tex](y + 1)(\frac{3(y + 3)}{y + 1} + 2) = (\frac{3y + 1}{y + 1})(y + 1) \\ \\ 3(y + 3) + 2y + 2 = 3y + 1[/tex]
That's better. But we still have have a parentheses that we need to simplify.
Use the distributive property.
[tex]3(y + 3) + 2y + 2 = 3y + 1 \\ \\
3y + 9 + 2y + 2 = 3y + 1[/tex]
Combine like terms.
[tex]3y + 9 + 2y + 2 = 3y + 1 \\ \\
5y + 11 = 3y + 1[/tex]
Subtract 3y from both sides of the equation.
[tex]5y + 11 - 3y = 3y + 1 - 3y \\ \\
2y + 11 = 1[/tex]
Subtract 11 from both sides of the equation.
[tex]2y + 11 - 11 = 1 - 11 \\ \\ 2y = -10[/tex]
Divide both sides by 2.
[tex]2y / 2= -10 / 2 \\ \\
y = -5[/tex]
So, y is equal to -5.
