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

2 Answers
Lucy
Apr 15, 2018

x=−3625
y=2125

Explanation:

3x−2y=−6 --- (1)
8x+3y=−9 --- (2)

From (1),

3x−2y=−6
3x=2y−6
x=23y−2 --- (3)

Sub (3) into (2)

8(23y−2)+3y=−9
163y−16+3y=−9
253y=7
y=2125 --- (4)

Sub (4) into (3)

x=23(2125)−2

x=−3625

ruthvera
Apr 15, 2018

you can use either elimination or substitution.

the answer is (−3625,2125)

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=−3625

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

3(−3625)−2y=−6

Once simplified, you should get y=2136
Your final answer is (−3625,2125)

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−63 <-- divide both sides by 3

x=(23)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(23y−2)+3y=−9 <-- this is what you got in the first step

3) 163y−16+3y=−9 <-- distrubute the 8

4) 253y=−9+16 <-- add like terms and then add more sides by 16

5)253y=7

6) y=7(325) <-- divide both sides by (25/3) which is the same thing as multiplying the reciprocal (3/25)

7) y=2125 <-- 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(2125)=−6

3) 3x−4225=−6

4) 3x=−6+4225

5) 3x=−10825

6) x=−10825⋅13

7) x=−3625 this is your x-value!

Your final answer is (−3625,2125)

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