Arrange the systems of equations that have a single solution in increasing order of the x-values in their solutions.

2x + y = 10

x - 3y = - 2

x + 2y = 5

2x + y = 4

5x + y = 33

x = 18 - 4y

y = 13 - 2x

8x + 4y = 52

x + 3y = 5

6x - y = 11

2x + y = 10

-6x - 3y = - 2

y = 10 + x

2x + 3y = 45

The order for the systems is 2), 5),7), 1) and 3)

Step-by-step explanation:

To find the solution for ecah system first you have to choose a method, for example you can isolate one of the variables (x or y) in both equations and then equal each other.

1) 2x + y = 10 ⇒ y = 10 - 2x

x - 3y = - 2 ⇒ - 3y = - 2 - x ⇒ y = (-2 - x) / -3 ⇒ y = 2/3 + (1/3) x

Now you have to equal both equations

10 - 2x = 2/3 + (1/3) x ⇒ 10 - 2/3 = (1/3) x + 2x ⇒28/3 = (7/3) x

⇒ (28/3) / (7/3) = x ⇒ 4.

With the "x" value obtained we return to the original equations and replace it there to find "y".

2x + y = 10 ⇒ 2 (4) + y = 10 ⇒ y = 10 - 8 ⇒ y = 2

The solution to the system is (4,2).

If you do the same for all the systems you will find that the solutions are

2) x + 2y = 5

2x + y = 4

Solution = (1, 2)

3) 5x + y = 33

x = 18 - 4y

Solution = (6,3)

4) y = 13 - 2x

8x + 4y = 52

Solution = ∅, no solution

5) x + 3y = 5

6x - y = 11

Solution = (2, 1)

6) 2x + y = 10

-6x - 3y = - 2

Solution = ∅, no solution.

7) y = 10 + x

2x + 3y = 45

Solution = (3, 13)

6, 2, 5, 7, 4, 1, 3

Step by Step:

1) x - 3 (10 - 2x) = - 2

x - 30 + 6x = - 2

7x - 30 = - 2

7x = 28

x = 4

2) x + 2 (4 - 2x) = 5

x + 8 - 4x = 5

-3x + 8 = 5

-3x = - 3

x = 1

3) 5 (18 - 4y) + y = 33

90 - 20y + y = 33

90 - 19y = 33

-19y = - 57

y = 3

x = 18 - 4 (3)

x = 18 - 12

x = 6

4) 8x + 4 (13 - 2x) = 52

8x + 52 - 8x = 52

52 = 52

5) x + 3 (-11 + 6x) = 5

x - 33 + 18x = 5

19x - 33 = 5

19x = 38

x = 2

6) - 6 (5 - 2y) - 3y = - 2

-30 + 12y - 3y = - 2

-30 + 9y = - 2

9y = 28

y = 3

2x + 3 (3) = 5

2x + 9 = 5

2x = - 4

x = - 2

7) 2x + 3 (10+x) = 45

2x + 30 + 3x = 45

5x + 30 = 45

5x = 15

x = 3