Solving equation where x is negative: why multiply by -1 for 3 - 7x = 5 ?

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clarkin

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Dear Colleagues,

I have the following equation: 3 - 7x = 5

Normally in order to isolate x, I would start by adding 3 to both sides and then dividing both sides by 7.

When I looked at the solution, it states that because the coefficient of x is negative, you need to multiply both sides of the equation by -1, this would make the -7 positive and positive 3, negative 3 and negative 5.
Therefore the equation becomes 7x - 3 = -5.

What I don't understand is why we need to multiply both sides of the equation by a negative number ? - what is the purpose of it?.

Thank you.
 
It is a sad state of affairs when an instructor tries to suggest there is ONLY ONE WAY to proceed.

This particular instructor apparently doesn't like negative numbers. The irony is that "multiply by -1" uses a negative number to solve that unnecessary fear. Also, what does this do to 3 and 5?

3 - 7x = 5

Part of the problem with this approach is the pathological desire to have the 'x' on the left. There is no need for that

Add 7x to both sides.

3 - 7x + 7x = 5 + 7x
3 = 5 + 7x

Whoops! That coefficient on x isn't negative any more.

Seriously, just learn to solve ANY WAY that comes to mind.

You will see this problem again with inequalities.

-7x > 12

You will be told, "Multiply or divide by a negative and you must reverse the inequality".

That is a fine and trusted rule, but why is that? Is it REALLY necessary?

You can avoid the rule by addition.

Add 7x

-7x + 7x > 12 + 7x

0 > 12 + 7x

Subtract 12

0 - 12 > 12 - 12 7x
-12 > 7x

We entirely avoided having to REMEMBER to reverse the inequality.

There are often MANY ways to proceed. Just pick one and proceed accurately.
 
Last edited:
clarkin said:
3 - 7x = 5

Normally in order to isolate x, I would start by adding 3 to both sides and then dividing both sides by 7.
Click to expand...
Let's see what happens, if we do that (first add 3 to each side and then divide each side by 7).

3 - 7x = 5

3 - 7x + 3 = 5 + 3

6 - 7x = 8

6/7 - 7x/7 = 8/7

6/7 - x = 8/7

Okay -- that's all valid, so far, but we have not yet isolated x. What next?


When I looked at the solution, it states that because the coefficient of x is negative, you need to multiply both sides of the equation by -1 … What I don't understand is why we need to multiply both sides of the equation by a negative number … what is the purpose of it?
Click to expand...
I disagree with the phrase "need to".

The author seems to imply that we must deal with a negative coefficient on x right away. That is not necessary.

If we solve an equation using familiar steps and we end up with something like:

-x = 2/7

then, as a final step, we simply multiply each side by -1 to get:

x = -2/7

In other words, it does not matter whether we multiply an equation by -1 at the very beginning, or in the middle, or at the very end of our work. What matters is that we follow the rules of algebra properly, and there's no rule in algebra that says we "need to" start solving certain equations by multiplying each side by -1.

Some instructors say things like, "You must do it this way, or I will take off points." If you ever hear something like this, then you must play the game according to their rules. (Later, you'll be free to choose your own steps). :cool:
 
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