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

2 Answers
Apr 15, 2018

Answer:

#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#

Apr 15, 2018

Answer:

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)#