Question

How do you solve `4x – y = 5` and `x + y = 10` and which method do you use?

Answer 1

`x=3` and `y = 7`

Explanation:

Here is the answer...

First take the second equation...
`x+y=10`
`x=10-y` ...`(eq. 1)`

Now substitute `(eq. 1)` in the given equation `4x - y = 5`.

`4 (10-y)-y = 5`

`40 - 4y - y = 5`

`40 - 5y = 5`

`40-5 = 5y`

`35 = 5y`

`y = 7`

Substituting this value of `y` in `(eq. 1)`, we get `x=3`.

Therefore, `x=3` and `y = 7`.

Answer 2

`(3,7)`

Explanation:

`4x-y=5--(1)`

`x+y=10--(2)`

solve by elimination

`(1)+(2)`

`5x=15`

`=>x=3`

substitute into `(2)`

`3+y=10`

`=>y=7`

check in `(1)`

`4xx3-7=12-7=5`

`=RHS`(1)#

`:.`consistent

`(3,7)`

Answer 3

See explanation.

Explanation:

The system is:

`{(4x-y=5),(x+y=10):}`

It can be solved using any of 3 methods:

• Using substitution:

From the second equation we can calculate that: `y=10-x`

If we put this in the first equation we get:

`4x-(10-x)=5`

`4x-10+x=5`

`5x-10=5`

`5x=15=>x=3`

Now we can calculate that `y=10-3=7`
So the solution is

`{(x=3),(y=7):}`

• By adding both equations:

In the initial system the coefficients of `y` are opposite numbers `-1` and `1`, so if we add both equations we get an equation with `x` variable only:

`5x=15`
`x=3`

Now we can calculate the remaining variable `y` by substitution:

`3+y=10=>y=7`

• Graphically

Both equations represent linear functions, so we can solve the system by graphing the lines and seeing if they intersect:

graph{(y-4x+5)(x+y-10)((x-3)^2+(y-7)^2-0.05)=0 [-10, 10, -8, 8]}

As we can see the lines intersect at `(3,7)`, so the solution is:

`{(x=3),(y=7):}`

The choice of method depends on the system of equations. Here the easiest (for me) is the second method but others may prefer different ones.