Question

Solve the following equation by Cramer’s method.

`("x" + "y" - 8)/2 = ("x" + 2y - 14)/3 = (3"x" - "y")/4`

Answer 1

`("x" + "y" - 8)/2 = ("x" + 2y - 14)/3`

3(x + y - 8) = 2(x + 2y - 14)

3x + 3y - 24 = 2x + 4y - 28

3x - 2x + 3y - 4y = -28 + 24 

1x - 1y = -4      ...(1)

`(x + 2y - 14)/3=(3x-y)/4`

4(x + 2y - 14) = 3(3x - y)

4x + 8y - 56 = 9x -3y

-56 = 9x - 4x - 3y - 8y

-56 = 5x - 11y

5x - 11y = -56     ...(2)

From 1 & 2

D = `|(1,-1),(5,-11)|=(1)(-11)-(5)(-1)`

= -11 - (-5)

= -11 + 5

= -6

D_x = `|(-4,-1),(-56,-11)|=(-4)(-11)-(-56(-1)`

= 44 - 56

= -12

D_y = `|(1,-4),(5,-56)|=(1)(-56)-(5)(-4)`

= -56 - (-20)

= -56 + 20

= -36

By Cramer’s Rule, we get

x = `D_x/D=-12/-6=2`

y = `D_y/D=-36/-6=6`

(x, y) = (2, 6) is the solution.

Answer 2

`("x" + "y" - 8)/2 = ("x" + 2y - 14)/3 = (3"x" - "y")/4`

Multiple by 4.

`(4("x" + "y" - 8))/2 = (4("x" + 2"y" - 14))/3 = (4(3"x" - "y"))/4`

Multiple by 3.

6(x + y − 8) = `(3 xx 4("x" + 2"y" - 14))/3` = 3(3x − y)

6x + 6y − 48 = 4x + 8y − 56 = 9x − 3y

6x + 6y − 48 = 9x − 3y

− 48 = 9x − 6x − 3y − 6y

− 48 = 3x − 9y

Dividing by 3.

− 16 = x − 3y       ...(1)

a₁ = 1, b₁ = −3, c₁ = −16

4x + 8y − 56 = 9x − 3y

− 56 = 9x − 4x − 3y − 8y

− 56 = 5x − 11y      ...(2)

a₂ = 5, b₂ = −11, c₂ = −56

By Cramer’s rule

D = `|("a"_1,"b"_1),("a"_2,"b"_2)|`

= `|(1,-3),(5,-11)|`

= 1 × (−11) − (−3) × 5

= −11 + 15

= 4

D_x = `|("c"_1,"b"_1),("c"_2,"b"_2)|`

= `|(-16,-3),(-56,-11)|`

= −16 × (−11) − (−3) × (−56)

= 176 − 168

= 8

D_y = `|("a"_1,"c"_1),("a"_2,"C"_2)|`

= `|(1,-16),(5,-56)|`

= 1 × (−56) − (−16) × 5

= −56 + 80

= 24

x = `"D"_"x"/"D" = 8/4` = 2

y = `"D"_"y"/"D" = 24/4` = 6

(x, y) = (2, 6)