Question

3x + 5y = 78
2x - y = 0

The point of intersection of the lines has an x-coordinate of _____.


78
6
-6

Please explain intersections to me, cause they make about zero sense.

Answer 1

you can multiply the 2nd equation times 5 and add the equations.

Answer 2

or, y=2x and sub

Answer 3

I was gonna say that

Answer 4

lol

Answer 5

I figured. I was kind of "proposing more complicated approach" (complicated, comparing to y=2x sub)

Answer 6

Dj, which was do you want to go ?
Substitution or elimination ?

Answer 7

The elimination sense to make more sense to me usually, but either works.

Answer 8

yes, so we will do substitution if elimination is something you comprehend. (To get all concepts going)

Answer 9

\(\large\color{black}{ \displaystyle 3x + 5y = 78 \\[0.4em] }\) (1st equation)
\(\large\color{black}{ \displaystyle 2x - y = 0 }\) (2nd equation)

Answer 10

Add y to both sides in the 2ND equation. What will your new (second) equation now be?

Answer 11

2x = 0+y, right?

Answer 12

yes, so that gives a second equation of
y=2x

(right ? )

Answer 13

OUR NEW SYSTEM NOW IS

\(\large\color{black}{ \displaystyle 3x + 5y = 78 \\[0.4em] }\) (1st equation)
\(\large\color{black}{ \displaystyle y=2x }\) (2nd equation)

Answer 14

Ok

Answer 15

Now, what does the second equation really say?
It tells you that [in this system of equations] "y" is the same thing
(or, 'has the same value as') as "2x".

that means that you can SUBSTITUTE "2x" instead of "y", into the 1st equation.

Answer 16

Our first equation.
\(\large\color{black}{ \displaystyle 3x + 5\color{red}{(y)} = 78 \\[0.4em] }\)

Substitution
\(\large\color{black}{ \displaystyle 3x + 5\color{red}{(2x)} = 78 \\[0.4em] }\)

Answer 17

is this substitution making sense ?

Answer 18

Not yet, but it's starting to I think.

Answer 19

sure, if you read of the replies and have any question, make sure to ask.

Answer 20

read **over** the replies....
(typo correction)

Answer 21

I'm still not getting it. It makes sense that 2x = y, but I don't get how to finish or understand it.

Answer 22

\(\large\color{black}{ \displaystyle y }\) is same thing as \(\large\color{black}{ \displaystyle 2x }\) (according to our 2nd equation)

So the first equation that says
\(\large\color{black}{ \displaystyle 3x+5\color{red}{y}=78 }\)

would be same exact thing as
\(\large\color{black}{ \displaystyle 3x+5(\color{red}{2x})=78 }\)

Answer 23

by doing this substitution we are not changing the value of the equation, just re-writing in the second term (the "5y") as "5 times 2x".

This allows us to find the X, but having to solve a simple equation with one variable.

Answer 24

Ok, that makes sense. So, how do you find x now?

Answer 25

well, you expand the \(\large\color{black}{ \displaystyle 5(\color{red}{2x}) }\) part, add all x's and on...

Answer 26

5(2x) => 10x

correct ?

Answer 27

Yeah, so that would leave 3x + 10x, which could be simplified to 13x?

Answer 28

yes

Answer 29

So you get
13x=78

Answer 30

and then x = ?

Answer 31

x = 6?

Answer 32

yup

Answer 33

Thank you so much!!! This makes so much sense now!

Answer 34

very nice.

we have found that x=6.... remember we said that y=2x ?

Answer 35

So, y = 12

Answer 36

yes

Answer 37

any questions ?

Answer 38

Nope, I think I've got it!! Thank you!!

Answer 39

You welcome !