Question

Solve the following system of equations by the elimination method:

`6x - 8/y = 14, 2x - 14/y = -1`

Answer 1

Given system of equations:

`6x - 8/y = 14`

`2x - 14/y = -1`

Step 1: Introduce new variables

Let `u = x` and `v = 1/y`.

Rewrite the system in terms of (u) and (v):

6u – 8v = 14 

2u – 14v = –1

Step 2: Eliminate one variable using the elimination method

Multiply the second equation by 3 to match the coefficient of (u) in the first equation:

6u – 8v = 14 

6u – 42v = –3

Subtract the second equation from the first:

(6u – 8v) – (6u – 42v) = 14 – (–3) 

6u – 8v – 6u + 42v = 14 + 3 

34v = 17 

`v = 17/34`

`v = 1/2`

Step 3: Substitute `(v = 1/2)` in one of the original equations to solve for (u)

Substitute in the second equation:

`2u - 14 xx 1/2 = -1` 

2u – 7 = –1 

2u = 6

u = 3

Step 4: Back-substitute (u) and (v) to find (x) and (y)

Recall: u = x = 3

`v = 1/y`

`v = 1/2`

⇒ y = 2