Question

How do you solve the system of equations `3x + 11y = 9` and `2x - 11y = 6`?

Answer 1

The solutions are `((x),(y))=((3),(0))`

Explanation:

We apply Cramer's rule

`3x+11y=9`

`2x-11y=6`

The determinants are

`D=|(3,11),(2,-11)|=3*(-11)-2*11=-55`

`D_x=|(9,11),(6,-11)|=9*(-11)-6*11=-165`

`D_y=|(3,9),(2,6)|=3*6-9*2=0`

The solutions are

`x=D_x/D=-165/-55=3`

`y=D_y/D=0/-55=0`

graph{(3x+11y-9)(2x-11y-6)=0 [-10, 10, -5, 5]}

Answer 2

`(x,y)to(3,0)`

Explanation:

`3xcolor(red)(+11y)=9to(1)`

`2xcolor(red)(-11y)=6to(2)`

`"note the terms in y have opposing signs so that adding"`
`"these terms together will eliminate y"`

`(1)+(2)" term by term"`

`(3x+2x)+(11y-11y)=(9+6)`

`rArr5x=15`

`"divide both sides by 5"`

`(cancel(5) x)/cancel(5)=15/5`

`rArrx=3`

`"substitute this value into either "(1)" or "(2)" and "`
`"solve for y"`

`"substituting in "(1)" gives"`

`(3xx3)+11y=9`

`rArr9+11y=9`

`"subtract 9 from both sides"`

`cancel(9)cancel(-9)+11y=9-9`

`rArr11y=0rArry=0`

`color(blue)"As a check"`

`"substitute these values in "(2)`

`(2xx3)-(11xx0)=6-0=6larr" True"`

`rArr"the point of intersection "=(3,0)`
graph{(y+3/11x-9/11)(y-2/11x+6/11)=0 [-10, 10, -5, 5]}