Question

Will Medal!

1:What is the solution set of the following equation?

-8x + x + 15 = -7x + 12

Ø
{1/7}
{all reals}

Answer 1

any ideas?

Answer 2

nope!!

Answer 3

Can you add like terms?

Answer 4

\(\color{#000000}{ \displaystyle -8x+x=~? }\)

Answer 5

i could do that but i don't know what x stands for!

Answer 6

\(\color{#000000}{ \displaystyle x }\) represents some unknown number that is said to satisfy/meet a certain equation.

Answer 7

So if I said, that I have 2 apples then I don't remember how many more apples I bought, and ended up with 5 apples.

Then, I can represent this as
\(2+x=5\)
(where \(x\) is just this unknown a mount of apples that I bought)

Answer 8

i get it

Answer 9

You don't want to write,

2 apples + "some unknown number of apples" = 5 apples

Instead you can shortly record

2 + x = 5

Answer 10

So, you have an equation,

\(\color{#000000}{ \displaystyle -8x + x + 15 = -7x + 12 }\)

(where \(x\) is just some unknown number)

Answer 11

And to solve this equation (and other equations similar to this one) you would use operations such as

1. Adding like terms
2. Adding/Subtracting a number or variable to both sides
3. Dividing both sides by a number

and such ....

Answer 12

basically, you want to simplify this equation (by performing the algebra that you know) to solve for \(x\).

Answer 13

So, let's actually solve this equation:
\(\color{#000000}{ \displaystyle -8x + x + 15 = -7x + 12 }\)

Answer 14

You see like terms on the left side (\(-8x\) and \(x\)).

Do you know how to add \(\color{#000000}{ \displaystyle -8x + x =~? }\)

Answer 15

sorry my laptop keeps freezing!!!

Answer 16

Yeah, I am also lagging. I am typically really fast with latex.

Answer 17

yes i think you first multiply -8 and x the the answer from that you add to x

like if x=2 it would be -8*2= (-16) then -16 + 2= -14 right??

Answer 18

Let me give you some similar example.
(I am frizzing a bit, so it will take more time)

Answer 19

Suppose you have the following equation,
\(\color{#000000}{ \displaystyle 3x+3 =7x-4x+3 }\)
and you want to determine the solution set
for \(x\) in this equation.

Just as,
(7 feet) - (4 feet) = (3 feet)
(7 computers) - (4 computers) = (3 computers)

so is,
\(\color{#000000}{ \displaystyle 7x-4x=3x }\)

(Essentially ..., 7 of something - 4 of something = 3 of this something.)

So, your equation now becomes
(instead of \(\color{#000000}{ \displaystyle 3x+3 =7x-4x+3 }\), into)

\(\color{#000000}{ \displaystyle 3x+3 =3x+3 }\)

and you can tell that in this case no matter what you plug in for \(x\) your equation is going to hold (since the sides are the same), therefore \(x\) can be any number, so the solution set is all real number.

Answer 20

What if I had (for example),
\(\color{#000000}{ \displaystyle 5x+7x-3 =12x+4 }\)

by adding like terms (since 5+7=12, therefore)
\(\color{#000000}{ \displaystyle 5x+7x=12x }\)

so you can re-write the equation as
\(\color{#000000}{ \displaystyle 12x-3 =12x+4 }\)

you can subtract 12x from both sides
\(\color{#000000}{ \displaystyle 12x-3 \color{red}{-12x}=12x+4 \color{red}{-12x} }\)

and since 12x-12x=0, therefore your equation will become,
\(\color{#000000}{ \displaystyle -3 =4 }\)

and when your system correctly simplifies to such a false statement that is obviously not true, you know that the equation from the very beginning could not have had any solutions.

So you don't have any solutions for \(x\) for which your initial equation is true, and therefore you have an empty set as your solutin set (because there are no solutions).

Answer 21

0=-3

Answer 22

Yes, indeed that is what your equation simplifies to.

Answer 23

okay i'm starting to get it!!

Answer 24

So, since your equation simplified to a false statement like that what can you say about the solution for \(x\) in this case ?

Answer 25

(I had an example where I ended with the false statement, and I concluded ... )

Answer 26

Wat do u mean

Answer 27

I was just hinting that when your equation simplifies
to false statements like 0=-3, -3=4, 7=111, then thi
implies that your equation does not have any solutions.

Answer 28

(No solutions is the same as an empty solution set)

Answer 29

oh

Answer 30

So, what is the solution set in your case?

Answer 31

wait can i make a guess for the answer?

Answer 32

Your equation was shown to be false (for any value of x) since you simplified it to get 0=-3. That means that the equation does not have any solution for \(x\), so your solution set would be empty - or, in other words, an empty solution set (denoted by Ø).

Answer 33

oh okay! so Ø is the answer?

Answer 34

Yes.

Answer 35

K thanks!