Solve the systems : 1) x = 3y 3x + y = 10 2)  2x + 7y = 1 2x - 2y = 9

1)

x = 3y

3x + y = 10

 From  the first equation, x = 3y. Substitute in the secon equation. We get: 3(3y)+3y = 10. So 9y+y = 10.  Or 10y = 10, y =1. Put y = 1in the 1st equation: x = 3.

(x,y) = (3,1) is the solution.

2)

 2x + 7y = 1

2x - 2y = 9

Subtract the  2nd equation from the 1st equation :

(2x+7y) - (2x-2y) = 1-9. X gets eliminated.

7y- -2y = 1-9 = -8

9y = -8, y = -8/9.

Put y = -8/9 and from the 2nd , 2x-2y = 9

2x-2(-8/9) = 9

2x= 9-16/9 = (81-16)/9

x = 65/18.

Therefore (x,y) = (65/18 , -8/9)

We'll solve the first system of equations using substitution technique:

x=3y

We'll substitute x by 3y, in the second equation of the system:

3*3y+y=10

We'll combine like terms:

9y+y=10

10y=10

We'll divide by 10 both sides:

y=1

Now, all we'll substitute the value for y into the first equation:

x=3y

x=3*1

x=3

The solution of the system is {(1 , 3)}.

Let's solve the second system of equations:

2x + 7y = 1

2x - 2y = 9

We'll solve this system using the substitution method:

2x=1-7y

Now, instead of 2x, we'll write in the second equation 1-7y.

1 - 7y - 2y =9

We'll combine like terms:

1 - 9y=9

We'll subtract 1 both sides:

-9y=8

We'll divide by -9 both sides:

y=-8/9

But 2x=1-7y and y=-8/9.

2x = 1 + 7*8/9

2x = 1 + 56/9

2x = (9+56)/9

2x = 65/9

We'll divide by 2:

x = 65/18

The solution of the system is: {(-8/9 , 65/18)}.