Question

How do you solve the following system?: `2x -5y =9 , 5x -3y = 2`

Answer 1

`y=-41/19color(white)(xxxx)x=-17/19`

Keeping in fraction form improves precision

Explanation:

Given:
`2x-5y=9..................................(1)`
`5x-3y=2..................................(2)`

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`color(blue)("Consider Equation (1)")`

Add `5y` to both sides
`2x=9+5y`

Subtract 9 from both sides.
`2x-9 =5y`

Divide both sides by 5
`2/5 x-9/5 =y`

Write in conventional form
`color(green)(y=2/5x-9/5....................................(1_a))`
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`color(blue)("Consider Equation (2)")`

Add `3y` to both sides
`5x=2+3y`

Subtract 2 from both sides
`5x-2=3y`

Divide both sides by 3
`5/3 x-2/3 = y`

Write in conventional form
`color(green)(y=5/3x-2/3....................................(2_a))`
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To end up with just 1 variable thus solvable.

`color(green)("Equation "(2_a))color(red)( =y= )color(blue)("Equation "(1_a)")`

`color(green)(2/5x-9/5) color(red)(=) color(blue)(5/3x-2/3)`

Collecting like terms
`2/5x -5/3x =9/5 -2/3`

`(6 -25)/15 x = (27 -10)/15`

`-19/15 x =17/15`

`-19x=17`

`color(red)(x=-17/19...........................(3))`
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Substitute for `x` in Equation (1)

`2x-5y=9..................................(1)`

`2color(red)((-17/19)) -5y=9`

`5y= 2(-17/19) -9`

`y= 2/5(-17/19)-9/5`

`color(red)(y=-2 3/19=-41/19...............(4))`