Verified answer

-2x + 2y = 6 can be written as:

-x + y = 3......(1)

8x - 7y = -4...(2)

so, 8x - 7(x + 3) = -4

=> x - 21 = -4

=> x = 17 => y = 20

:)>

This system of equation can be solved in two ways :

METHOD 1 :

from first equation -2x + 2y = 6

=> -x + y = 3

=> y - x = 3

=> y = x + 3 ------------- Name this equation 1.

Now, from the second equation, you have

8x - 7y = - 4

now in the place of y, we can put x + 3, since from 1 above we had calculated y = x + 3.

so now the second equation becomes

8x - 7(x+3) = - 4

8x - 7x - 21 = - 4

x - 21 = - 4

=> x = 21 - 4 = 17

since from 1 we have y = x + 3, so y = 17 + 3 = 20.

so x = 17 and y = 20.

METHOD 2 :

from the first equation we have

- 2x + 2y = 6

2y - 2x = 6

multiplying this whole equation by 4, we have

4(2y) - 4(2x) = 4(6)

or 8y - 8x = 24. --------------- Name this A.

also from the second equation we have

8x - 7y = - 4. ------------ Name this B.

On adding A and B,

we get

8y - 8x + 8x - 7y = 24 - 4

=> 8y - 7y = 20

=> y = 20.

Plugging this into A,( you could plug this into B, but oh well)

we have

8y - 8x = 24

=> 8(20) - 8x = 24

=> 160 - 8x = 24

=> 8x = 136

=> x = 17

So we get x = 17 and y = 20.

CHECK

from first equation, - 2x + 2y = 6.

=> -2(17) + 2(20) = 6

=> -34 + 40 = 6 or 6=6 which is true.

from second equation,

8x - 7y = -4

=> 8(17) - 7(20) = - 4

=> 136 - 140 = - 4 or - 4 = - 4 which is true.

So both equations are simultaneously satisfied. Thus the solutions we got are correct. (PHEW !!!)

Thus x = 17 and y = 20. ( YAY !!!)

EDIT : SECOND QUESTION

METHOD 1

From First Equation

3x - 8y = 0

=> 3x = 8y

=> x = 8y/3 ------------- Name this D.

from second equation

2x - 3y = -14

2(8y/3) - 3y = - 14 ------------- from D, we have x = 8y/3. so in place of x, I can write 8y/3.

=>16y/3 - 3y = - 14

=> 16y - 9y = - 42

=> 7y = - 42

=> y = - 6

also we calculated x = 8y/3 ------------ from D

x = 8(-6)/3 = - 48/3 = - 16

METHOD 2

we have

3x - 8y = 0

2x - 3y = -14

Multiplying first equation by 8, we have

24x - 64y = 0 -------- Name this E.

Multiplying the second equation by 12, we have

24x - 36y = -168 ------------- Name this F.

Now, subtracting F from E, we have

24x - 36y - 24x + 64y = -168

=> 64y - 36y = -168

=> 28y = -168

=> y = -168/28 = -6.

Substituting in either of the equations( the original equations, E or F), we get the value of x as -16.

Thus x = -16, y = - 6.

CHECK :

first equation --->

3(-16) - 8(-6) = 0

-48 + 48 = 0 or 0 = 0, which is true.

second equation ---->

2(-16) - 3(-6) = -14

-32 + 18 = -14 or -14 = -14, which is true.

Thus the values which we found out are correct as the equations are simultaneously satisfied.

Thus x = -16, y = -6.

Hope that helped.

Regards

Sid

(1) : - 2x + 2y = 6 → (1') : - x + y = 3

(2) : 8x - 7y = - 4

You calculate (2) + [4 * (1)]

(8x - 7y) + 4(- 2x + 2y) = - 4 + (2 * 6)

8x - 7y - 8x + 8y = - 4 + 12

→ y = 8

Then you substitute y by 8 into the equation (1')

- x + y = 3

- x + 8 = 3

- x = - 5

→ x = 5