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

Apr 15, 2018

$x = - \frac{36}{25}$
$y = \frac{21}{25}$

Explanation:

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

From (1),

$3 x - 2 y = - 6$
$3 x = 2 y - 6$
$x = \frac{2}{3} y - 2$ --- (3)

Sub (3) into (2)

$8 \left(\frac{2}{3} y - 2\right) + 3 y = - 9$
$\frac{16}{3} y - 16 + 3 y = - 9$
$\frac{25}{3} y = 7$
$y = \frac{21}{25}$ --- (4)

Sub (4) into (3)

$x = \frac{2}{3} \left(\frac{21}{25}\right) - 2$

$x = - \frac{36}{25}$

Apr 15, 2018

you can use either elimination or substitution.

the answer is $\left(- \frac{36}{25} , \frac{21}{25}\right)$

Explanation:

WAY 1) Elimination

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

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

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.

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

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

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.

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

$25 x = - 36$

$x = - \frac{36}{25}$

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

$3 \left(- \frac{36}{25}\right) - 2 y = - 6$

Once simplified, you should get $y = \frac{21}{36}$
Your final answer is $\left(- \frac{36}{25} , \frac{21}{25}\right)$

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 $3 x - 2 y = - 6$. You could also solve for x in the other equation, or solve for y, it's really up to you!

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

$x = \frac{2 y - 6}{3}$ <-- divide both sides by 3

$x = \left(\frac{2}{3}\right) y - 2$ <-- simplify.

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

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

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

2) $8 \left(\frac{2}{3} y - 2\right) + 3 y = - 9$ <-- this is what you got in the first step

3) $\frac{16}{3} y - 16 + 3 y = - 9$ <-- distrubute the 8

4) $\frac{25}{3} y = - 9 + 16$ <-- add like terms and then add more sides by 16

5)$\frac{25}{3} y = 7$

6) $y = 7 \left(\frac{3}{25}\right)$ <-- divide both sides by (25/3) which is the same thing as multiplying the reciprocal (3/25)

7) $y = \frac{21}{25}$ <-- this is your y value!

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

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

2) $3 x - 2 \left(\frac{21}{25}\right) = - 6$

3) $3 x - \frac{42}{25} = - 6$

4) $3 x = - 6 + \frac{42}{25}$

5) $3 x = - \frac{108}{25}$

6) $x = - \frac{108}{25} \cdot \frac{1}{3}$

7) $x = - \frac{36}{25}$ this is your x-value!

Your final answer is $\left(- \frac{36}{25} , \frac{21}{25}\right)$