SOLVING SYSTEMS OF EQUATIONS BY ELIMINATION

Problem 1 :

3x + ky = 8

x + 4y = -1

If (x, y) is a solution to the system of equations above and k is constant, what is y in terms of k ?

(a)  5/(k - 12)     (b)  7/(k -4)     (c) 11/(k - 12)      (d)  9/(k -4)

Solution :

3x + ky = 8   ---(1)

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

To solve for y, we have to eliminate x.

Coefficient of x in 1^st and 2^nd equations are 3 and 1 respectively.

Eliminating x :

Multiply the 2^nd equation by 3 and subtract from (1)

3x + ky - 3(x + 4y) = 8 - 3(-1)

3x + ky - 3x - 12y = 8 + 3

y(k-12) = 11

Divide by (k - 12) on both sides, we get

y = 11/(k - 12)

Problem 2 :

x/(y +2) = 2

3(y - 5) - x = -16

If (x, y) is the solution to the system of equations above, what is the value of x ?

Solution :

Coefficients of x in (1) and (2) are same. To eliminate x we add (1) and (2).

(1) + (2)

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

0x - 2y - x + 3y = 4 - 1

y = 3

By applying the value of y (1), we get

x - 2(3) = 4

x - 6 = 4

x = 4 + 6

x = 10

So, the value of x is 10.

Problem 3 :

-2x - y = -9

5x - 2y = 18

Which of the following ordered pairs (x, y) fulfills the system of equations above ?

(a)  (-4, 1)    (b)  (2, 5)    (c)  (3, 3)     (d)  (4, 1)

Solution :

-2x - y = -9 ------(1)

5x - 2y = 18 ------(2)

Coefficients of y in (1) and (2) are -1 and -2. To eliminate y, we have to multiply the first equation by 2 and subtract.

2(-2x-y) - (5x - 2y) = 2(-9) - 18

-4x - 2y - 5x + 2y = -18 - 18

-9x = -36

Divide by -9 on both sides, we get

x = 4

By applying the value of x in (1)

-2(4) - y = -9

-8 - y = -9

Add 8 on both sides.

-y = -9 + 8

-y = -1

y = 1

So, the required solution is (4, 1).

Problem 4 :

-3x + 2y = 5

-9x + 6y = 18

The system of equations above has how many solutions (x, y) ?

(a)  Zero     (b)  One     (c)  Two      (d)  More than two

Solution :

-3x + 2y = 5  -----(1)

-9x + 6y = 18 -----(2)

Coefficients of y in (1) and (2) are -3 and -9. To eliminate y, we have to multiply the first equation by 3 and subtract.

3(1) - (2)

3(-3x + 2y) - (-9x + 6y) = 3(5) - 18

-9x + 6y + 9x - 6y = 15 - 18

0 ≠ -3

No values of x and y is not satisfying the equation, so there is no solution. Zero solution.

Problem 5 :

(1/3) x + (1/6) y = 5

(3/5) x + (1/5) y = -4

Which of the following ordered pairs (x, y) fulfills the system of equations above ?

(a)  (-50, 130)     (b)  (2, 26)    (c)  (5, 20)      (d)  (20, -10)

Solution :

(1/3) x + (1/6) y = 5  ----(1)

(3/5) x + (1/5) y = -4 ----(2)

2x + y = 30 ----(3)

3x + y = -20 ----(4)

(3) - (4)

2x + y - (3x + y) = 30 - (- 20)

2x + y - 3x - y = 30 + 20

-x = 50

x = -50

By applying the value of x in (1), we get

(1/3)(-50) + (1/6) y = 5 

y/6 = 5 + 50/3

y/6 = 65/3

Multiplying by 6 on both sides, we get

y = (65/3) 6

y = 130