Question

Solve the system of equations algebraically.

4x-3y=6
6x-5y=6


a. (5,7)
b. many solutions
c. no soluton
d. (6,6)

Answer 1

Option (d) is correct.

(6, 6) is the solution of the given system of equation.

Explanation:

Consider the given equations

4x-3y=6 ....(1)

6x-5y=6 .........(2)

Solving system of equation using elimination method,

Multiply equation (1) by 5 , we get

(1) ⇒ 20x - 15y = 30 ......(3)

Multiply equation (2) by 3 , we get,

(2) ⇒ 18x - 15y = 18 .......(4)

Subtract equation (3) from (4) , we get,

18x - 15y -( 20x - 15y) = 18 - 30

18x - 15y - 20x +15y = -12

18x - 20x = -12

-2x = -12

⇒ x = 6

Also put x = 6 in (1) , we get

(1) ⇒ 4x - 3y = 6

⇒ 4(6) - 3y = 6

⇒ 24 - 3y = 6

⇒ - 3y = 6-24

⇒ - 3y = - 18

⇒ y = 6

Thus, Option (d) is correct.

(6, 6) is the solution of the given system of equation.

Answer 2

Option d. (6,6)

Explanation:

(1) 4x-3y=6

(2) 6x-5y=6

Using the equal values method. Let's isolate "x" from each equation:

(1) 4x-3y=6

Adding 3y to both sides of the equation:

4x-3y+3y=6+3y

4x=6+3y

Dividing both sides of the equation by 4:

`(4x)/(4)=(6+3y)/(4)\\ x=(6+3y)/(4)`

Isolating "x" from the second equation:

(2) 6x-5y=6

Adding 5y to both sides of the equation:

6x-5y+5y=6+5y

6x=6+5y

Dividing both sides of the equation by 6:

`(6x)/(6)=(6+5y)/(6)\\ x=(6+5y)/(6)`

Equaling y:

y=y

`(6+3y)/(4)=(6+5y)/(6)`

Solving for "y": Cross multiplication:

6(6+3y)=4(6+5y)

Applying the distributive property:

6(6)+6(3y)=4(6)+4(5y)

36+18y=24+20y

Subtracting from both sides 18y and 24:

36+18y-18y-24=24+20y-18y-24

12=2y

Dividing both sides by 2:

12/2=2y/2

6=y

y=6

Replacing y by 6 in the any of the equations where we isolated "x":

`x=(6+3y)/(4)\\ x=(6+3(6))/(4)\\ x=(6+18)/(4)\\ x=(24)/(4)\\ x=6`

The solution is (x,y)=(6,6)