Question

Solve the following system of equations `[((1) " ", (2x + y + 3z = 12)),((2) " ",(4y – z = -7)),((3) " ",(5x + 8z = 34))]`?

Answer 1

`x=2`
`y=-1`
`z=3`

Explanation:

`color(lightgreen)(( 1 )` `color(lightgreen)rarr` `2x+y+3z=12`
`color(lightblue)(( 2)` `color(lightblue)rarr` `4y-z=-7`
`color(pink)(( 3)` `color(pink)rarr` `5x+8z=34`

Using `color(lightblue)((2)`:
`color(orange)((4)` `color(orange)rarr` `z=4y+7`

Substituting the value of `z` in `color(pink)((3)`:
`5x+8(4y+7)=34`

`5x+32y+56=34`

`color(blue)((5)` `color(blue)rarr` `5x+32y=-22`

Substituting value of `z` in `color(lightgreen)((1)`:
`2x+y+3(4y+7)=12`

`2x+y+12y+21=12`

`color(purple)((6)` `color(purple)rarr` `2x+13y=-9`

Solving `color(blue)((5)` & `color(purple)((6)` simultaneously:
`5x+32y=-22`
`2x+13y=-9`

Multiply `color(purple)((6)` by `-2.5`:
`cancel(5x)+32y=-22`
`cancel(-5x)-32.5y=+22.5`

`-0.5y=0.5`

`color(red)(y=-1)`

Substituting value of `color(red)y` in `color(purple)((6)`:
`2x+13(-1)=-9`

`2x-13=-9`

`2x=4`

`color(red)(x=2)`

Substituting value of `color(red)y` in `color(orange)((4)`:
`z=4(-1)+7`

`z=-4+7`

`color(red)(z=3)`