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$y = 4y + 9$

How do I isolate y?

Can I do

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$y = 4y + 9$

$\frac{y}{4y} = 9$ etc

Also some other questions please:

- $\frac{5x + 1}{3} – 4 = 5 – 7x$

In the above (1), if I want to remove the ‘3’ from the denominator of the LHS, do I multiply **everything** on the RHS by 3? What about the ‘-4’ on the LHS, do I do anything to that?

- In the above

$\frac{y}{4y}$

If I want to solve it, do I cross a y out from the top, or from the bottom? Does that make sense ? :\

Thanks in advance..I have an exam tomorrow..

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To make $y$ the subject or isolate $y$ try to think of equations as ‘weighing scales’ you can carry out any operation you like but you must do the same to both sides (to keep it balanced):

$$y = 4y + 9$$

$$\implies y-4y=9+4y-4y\tag{1}$$

$$\implies-3y=9$$

$$\implies \frac{-3y}{-3}=\frac{9}{-3}\tag{2}$$

$$\implies y =-3$$

For $(1)$ I subtracted $4y$ from *both sides*.

For $(2)$ I divided *both sides* by $-3$.

To move on

$$\cfrac{5x+1}{3}-4=5-7x$$

$$\implies \cfrac{5x+1}{3}=9-7x$$

$$\implies 5x+1=3\times(9-7x)$$

$$\implies 5x+1=27-21x$$

$$\implies 26x+1=27$$

$$\implies 26x=27-1$$

$$\implies 26x=26$$

$$\implies x=1$$

Others have already given you the answer, so I won’t reiterate.

Instead let me point out to you how you should think about such questions.

It appears to me that you have no firm grasp of what equations are and how to manipulate them. This is not a shame, it’s not easy to learn and also not easy to teach.

You’re approaching the problem with the mindset “here is the equation, what technique do I need to use?”. People trained in mathematics use techniques without thinking much about them, because it has become second nature. However, when you learn about the techniques you should never try to blindly apply techniques without understanding. At best, this leads to a lot of mental overhead because you have to memorize seemingly “random” facts. At worst, it leads to general confusion and to you applying methods when they seem to work but actually don’t.

Think of the equation as a scale. Everything you do to one side, you also want to do to the other side. If you compute $\frac{y}{4y}$, you effectively divide the left hand side by $4y$. But does that match up what you’re doing to the right hand side? Assuming you have a firm grasp of how to manipulate fractions, you should see that this is not the case.

I hope this helps and doesn’t come across as too condescending, I could be judging your level of knowledge wrong.

You have the right idea to get $y$ by itself on the left. However, instead of dividing by $4y$, you would subtract $4y$ from both sides. This will give you

$$y – 4y = 9.$$

Now, if you combine the like $y$-terms on the left. What would $y – 4y$ turn into? You can think of this as: $1y – 4y$.

For your next problem, if you want to multiply both sides by $3$, then you multiply **everything** by $3$, including the $-4$ on the left-hand side.

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