Question

How do you solve this system of equations `\frac { 2x + y } { 3} = 15;\frac { 3x - y } { 5} = 1`?

Answer 1

`(x,y)to(10,25)`

Explanation:

`"the first step is to eliminate the fraction in both equations"`

`"multiply both sides by 3 for "(2x+y)/3=15`

`cancel(3)xx(2x+y)/cancel(3)=3xx15`

`rArr2x+y=45`

`"multiply both sides by 5 for "(3x-y)/5=1`

`cancel(5)xx(3x-y)/cancel(5)=5xx1`

`rArr3x-y=5`

`"the two equations to be solved are therefore"`

`2x+color(red)(+y)=45to(1)`

`3xcolor(red)(-y)=5to(2)`

`"we can eliminate the term in y by adding the equations"`

`(1)+(2)" term by term on both sides"`

`(2x+3x)+(y-y)=(45+5)`

`rArr5x=50`

`"divide both sides by 5"`

`rArrx=10`

`"substitute this value into either of the 2 equations"`
`"and solve for y"`

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

`(2xx10)+y=45`

`rArry=45-20=25`

`color(blue)"As a check"`

`"substitute these values in "(2)`

`(3xx10)-25=30-25=5larr" True"`

`rArr" point of intersection "=(10,25)`
graph{(y+2x-45)(y-3x+5)((x-10)^2+(y-25)^2+0.02)=0 [-80, 80, -40, 40]}