Solve the systems :

1)

x = 3y

3x + y = 10

2)

2x + 7y = 1

2x - 2y = 9

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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)}.**

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

3x + y = 10.........(2)

Using the substitution method, we will substitute x values from (1) in equation (2):

==> 3x + y = 10

==> 3(3y) + y = 10

==> 9x + y = 10

==> 10y = 10

**==> y= 1**

==> x= 3y = 3*1 = 3

**==> x= 3**

2x + 7y = 1........(1)

2x - 2y = 9.........(2)

Using the elimination method, we will subtract (2) from (1):

==> 9y = -8

**==> y= -8/9**

Now to find x value , we will substitute with with either (1) or (2):

==> 2x - 2y = 9

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

==> 2x + 16/9 = 9

==> 2x = 9 - 16/9

==> 2x = 65/9

**==> x= 65/18**

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)

For the system of equations

x=3y and 3x+y=10

3x+y=10

=>9y+y=10

=>10y=10

=>y=1

=>x=y = 3

**Therefore for the first system x=1 and y=3**

For the second system of equations: 2x+7y=1 and 2x-2y=9

Subtract the 2nd from the first, we get 7y+2y=1-9

=>9y=-8

=>y=-8/9

Substitute this is 2x-2y=9

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

=>2x+16/9=9

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

=>x=(65/18)

**Therefore for this system of equations x=65/18 and y=-8/9**

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