Question

What is the solution to the above system of equations? (I'll post below)

Answer 1

Options

Answer 2

Can somebody please help? What do I do first?

Answer 3

\[9x-2y=11\]\[5x-2y=15\]

Look at the two equations. Do you see any variables where the coefficients are either equal or equal but opposite in sign?

Answer 4

the coefficients are equal right? @whpalmer4

Answer 5

Some of them are. Which variables do the equal ones go with?

Answer 6

9 and 5 go together and 2 and 2, right?

Answer 7

@whpalmer4

Answer 8

Which variables? Variables are represented by letters such as \(x\) and \(y\). Which variable has coefficients which are equal in both equations?

Answer 9

x and y?

Answer 10

Does 9 = 5?

Answer 11

no

Answer 12

Okay. Then the variable associated with 9 and 5 does not have equal coefficients in both equations. Which variable is associated with 9 and 5?

Answer 13

x

Answer 14

x since it is 9x and 5x right?

Answer 15

right. so which variable does have coefficients which are equal?

Answer 16

y!

Answer 17

or 2

Answer 18

So now what?

Answer 19

\(y\) is the variable. 2 (or -2, actually) is the coefficient.

Answer 20

ok

Answer 21

Perhaps you didn't understand the terminology. Now you do :-)

Answer 22

So now that I know that...what is the next step

Answer 23

So, our goal is to get rid of one of the two variables, giving us an equation in just one variable, which we can solve in the usual fashion.

We have two choices here: we can multiply one of the equations by -1, then add the two equations together, OR,
we can subtract one equation from the other.

I prefer the first approach, because I find subtracting polynomials to be a bit more error-prone for many than adding polynomials.

Your choice: pick one of the polynomials and multiply each term by -1. What is the result you get?

Answer 24

I guess I'll do the first one to clear out from the errors

Answer 25

So do I multiply both sides by -1?

For example: 9 x -2y (-1) = 11 (-1)?

Answer 26

You multiply the entire equation by -1:

\[3a+4b = 5\]\[(-1)*3a + (-1)*4b = (-1)*5\]\[-3a-4b=-5\]

Answer 27

You just change the signs of each term...

Answer 28

-9x + 2y = -11 ? I changed the 2y into a positive since it was already a negative. Is this right?

Answer 29

Yes. Very good. Now, add the two equations together:

\[-9x+2y=-11\]\[~~~5x-2y=~~~15\]-----------------

What do you get?

Answer 30

-4x = 4 ?

Answer 31

is that right?

Answer 32

Yes. Can you solve that for \(x\)?

Answer 33

This procedure is called elimination, because we have eliminated one of the variables.

Answer 34

@whpalmer4

Answer 35

Hello, can you solve \(-4x=4\) for \(x\)?

Answer 36

Sorry I left but I am back now. Are you willing to finish this?
And as for elimination, does this mean I have to divide both sides by four?

Answer 37

@whpalmer4

Answer 38

Hi, now I'm back, and you're gone :-)

No, the term "elimination" comes from the fact that we "eliminated" \(y\) as a variable from our equations.

To solve \[-4x=4\]we would divide both sides by \(-4\), not \(4\). When you do so, you get \[\frac{-4x}{-4} = \frac{4}{-4} \]\[x = -1\]

Now we take \(x=-1\) and plug it back into either of our initial equations and solve for \(y\).

\[9x-2y=11\]\[9(-1)-2y=11\]\[-9-2y=11\]\[-2y=20\]\[y=-10\]

And (very important!) we check the solution in ALL of the equations:

\[9x-2y=11\]\[9(-1)-2(-10) = 11\]\[-9+20=11\]\[11=11\checkmark\]

\[5x-2y=15\]\[5(-1)-2(-10)=15\]\[-5+20=15\]\[15=15\checkmark\]

So the solution to this system of equations is \[x=-1,\,y=-10\] or \((-1,-10)\)

If we graph the two equations, we'll see that they intersect at \((-1,-10)\):