SOLVE THE SYSTEM OF LINEAR EQUATIONS BY ELIMINATION

Problem 1 :

3x + y = 7

-2x - y = 9

Solution :

3x + y = 7 ---> (1)

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

Add (1) + (2), we get

3x + y - 2x - y = 7 + 9

x = 16

By applying x = 16 in (1) equation, we get

3(16) + y = 7

48 + y = 7

y = 7 - 48

y = -41

Therefore, the solution is x = 16 and y = -41.

Problem 2 :

4x + 3y = 2

2x - 3y = 1

Solution :

4x + 3y = 2 ---> (1)

2x - 3y = 1 ---> (2)

Add (1) + (2), we get

4x + 3y + 2x - 3y = 2 + 1

6x = 3

x = 1/2

By applying x = 1/2 in (1) equation, we get

4(1/2) + 3y = 2

2 + 3y = 2

3y = 2 - 2

3y = 0

y = 0

Therefore, the  solution is x = 1/2 and y = 0.

Problem 3 :

2x + 2y = 3

x = 4y - 1

Solution :

2x + 2y = 3 ---> (1)

x - 4y = -1 ---> (2)

(1) ∙ 2 ==> 4x + 4y = 6 ----> (3)

Add (3) + (2), we get

4x + 4y + x - 4y = 6 - 1

5x = 5

x = 1

By applying x = 1 in (1) equation, we get

2(1) + 2y = 3

2 + 2y = 3

2y = 3 - 2

2y = 1

y = 1/2

Therefore, the solution is x = 1 and y = 1/2.

Problem 4 :

y = 1 + x

2x + y = -2

Solution :

y = 1 + x

x - y = -1 ---> (1)

2x + y = -2 ---> (2)

Add (1) + (2), we get

x - y + 2x + y = -1 - 2

3x = -3

x = -1

By applying x = -1 in equation (1)

y = 1 - 1

y = 0

Therefore, the solution is x = -1 and y = 0.

Problem 5 :

1/2x + 4y = 4

2x - y = 1

Solution :

1/2x + 4y = 4 ---> (1)

2x - y = 1 ---> (2)

(1) ∙ 4 ==> 8x - 4y = 4 ---> (3)

Add (1) + (3), we get

1/2x + 4y + 8x - 4y = 4 + 4

17/2x = 8

17x = 16

x = 16/17

By applying x = 16/17 in equation (2)

2(16/17) - y = 1

32/17 - y = 1

-y = (17 - 32) / 17

-y = -15/17

y = 15/17

Therefore, the solution is x = 16/17 and y = 15/17.

Problem 6 :

y = x - 4

4x + y = 26

Solution :

y = x - 4

x - y = 4 ---> (1)

4x + y = 26 ---> (2)

Add (1) + (2), we get

x - y + 4x + y = 4 + 26

5x = 30

x = 6

By applying x = 6 in equation (1)

y = 6 - 4

y = 2

Therefore, the solution is x = 6 and y = 2.

Problem 7 :

3x + 4y = -23

2y - x = -19

Waht is the solution (x, y) to the system of eqautions above ?

a)  (-5, -2)   b)  (3, -8)     c)  (4, -6)    d) (9, -6)

Solution :

3x + 4y = -23 ----(1)

- x + 2y = -19 -----(2)

(1) + 3(2)

3x + 4y - 3x + 6y = -23 - 57

10y = -80

y = -80/10

y = -8

Applying the value of y, we get

2(-8) - x = -19

-16 - x = - 19

-x = -19 + 16

-x = -3

x = 3

So, the required point is (3, -8). Option b is correct.

Problem 8 :

b = 2.35 + 0.25x

c = 1.75 + 0.40x

In the equation above b and c represent the price per pound in dollars, of beef and chicken respectively, x weeks after July 1 during last summer. What was the price per pound of beef when it was equal to the price per pound of chicken ?

Solution :

b = 2.35 + 0.25x

c = 1.75 + 0.40x

b = c

2.35 + 0.25x = 1.75 + 0.40x

2.35 - 1.75 = 0.40x - 0.25x

0.6 = 0.15x

x = 0.6/0.15

x = 4

Cost of beef per pound = ?

Applying x = 4, we get

b = 2.35 + 0.25(4)

= 2.35 + 1

= $3.35