Question

How do solve the following linear system?: `3x - 2y = -6 , 8x + 3y= -9`?

Answer 1

`x=-36/25`
`y=21/25`

Explanation:

`3x-2y=-6` --- (1)
`8x+3y=-9` --- (2)

From (1),

`3x-2y=-6`
`3x=2y-6`
`x=2/3y-2` --- (3)

Sub (3) into (2)

`8(2/3y-2)+3y=-9`
`16/3y-16+3y=-9`
`25/3y=7`
`y=21/25` --- (4)

Sub (4) into (3)

`x=2/3(21/25)-2`

`x=-36/25`

Answer 2

you can use either elimination or substitution.

the answer is `(-36/25, 21/25)`

Explanation:

WAY 1) Elimination

Take you two equations and line them up horizontally like so:

`3x-2y=-6`
`8x+3y=-9`

Check to see if the x coefficients of the two equations are the same or if the y coefficients are the same. In this case, they are not. So you'll have to multiply both equations by a common factor to either make the y coefficients or the x coefficients be the same. I decided to make the y coefficients the same.

In order to do that, multiply the whole equation by the least common multiple of the y coefficients. So our y coefficients of the two equations are -2 and 3. The LCM of the two numbers is 6. So multiply both the equations by 6.

`3(3x-2y=-6)` <-- multiply by 3 to make the y coefficient equal 6
`2 (8x+3y=-9)` <-- multiply by 2 to make the y coefficient equal 6

`9x-6y=-18`
`16x+6y=-18`

Notice that now you can add the two equations together to get rid of the y coefficients completely, in other words, you're eliminating it.

`9x-6y=-18`
+`16x+6y=-18`

`25x=-36`

`x=-36/25`

THIS IS YOUR X VALUE! Now plug in your x value into either of your equations to solve for the y value.

`3(-36/25)-2y=-6`

Once simplified, you should get `y= 21/36`
Your final answer is `(-36/25, 21/25)`

WAY 2) Substitution

Solve for a variable in one equation and then substitute that into either the same equation or the other equation given.

STEP 1: For this problem, I decided to solve for x in the equation `3x-2y=-6`. You could also solve for x in the other equation, or solve for y, it's really up to you!

`3x-2y=-6`
`3x=2y-6` <-- add 2y to both sides

`x=(2y-6)/3` <-- divide both sides by 3

`x=(2/3)y-2` <-- simplify.

STEP 2: Now plug in that what you get as your answer as x into either one of your equations! (you could use `3x-2y=-6` or `8x+3y=-9`) i decided to use `8x+3y=-9` but you could use any.

So plug in the x into the equation of your choice:

1) `8x+3y=-9`

2) `8(2/3y-2)+3y=-9` <-- this is what you got in the first step

3) `16/3y-16+3y=-9` <-- distrubute the 8

4) `25/3y=-9+16` <-- add like terms and then add more sides by 16

5)`25/3y=7`

6) `y=7(3/25)` <-- divide both sides by (25/3) which is the same thing as multiplying the reciprocal (3/25)

7) `y= 21/25` <-- this is your y value!

STEP 3 plug the y value you just found into either one of the equations. I chose the `3x-2y=-6` equation but it doesn't matter which one you pick!

1) `3x-2y=-6`

2) `3x-2(21/25)=-6`

3) `3x-42/25=-6`

4) `3x=-6+42/25`

5) `3x=-108/25`

6) `x = -108/25 * 1/3`

7) `x=-36/25` this is your x-value!

Your final answer is `(-36/25, 21/25)`