Question

How do I use elimination to find the solution of the system of equations `y=1/3x+7/3` and `y=−5/4x+11/4`?

Answer 1

You multiply each equation by the LCM of the denominators and then solve by elimination as usual.

Explanation:

(1) `y = 1/3x+7/3`
(2) `y = -5/4x + 11/4`

Step 1: Multiply each equation by the least common multiple (LCM) of the denominator.

Multiply Equation 1 by `3` and Equation 2 by `4`.

(3) `3y = x+7`
(4) `4y = -5x + 11`

Step 2: To solve by elimination, multiply Equation 3 by `5`.

(5) `15y = 5x+35`
(4) `4y = -5x + 11`

Step 3: Add the two equations.

`19y = 46`

(6) `y= 46/19`

Step 4: Substitute Equation 6 into Equation 3.

`x+7 = 3y = 3×46/19 = 138/19`

`x= 138/19 -7 = (138-133)/19 = 5/19`

The solution is `x = 5/19`, `y = 46/19`

Check:

Substitute in Equation 1.

`y = 1/3x+7/3`

`46/19 = 1/3× 5/19 + 7/3 = 5/57 + 7/3 = 5/57 + 133/57 =138/57 =46/19`

Substitute in Equation 2.

`y = -5/4x + 11/4`

`46/19 = -5/4× 5/19 + 11/4 = -25/76 + 11/4 = -25/76 + 209/76 = 184/76 = 46/19`