Question

The sum of two numbers is 22, and their difference is 12. What are the numbers?

Answer 1

`rarrx=17`

`rarry=5`

Explanation:

Let the numbers be `xand y`

Then,

`color(blue)(x+y=22`

`color(blue)(x-y=12`

We can solve the problem using substitution

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the first equation,

`rarrx+y=22`

`rarrx+y-color(red)(y)=22-color(red)(y)`

`rarrx=22-y`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now, we know that `x` is `(22-y)`. So, substitute the value to the second equation

`rarrx-y=12`

`rarr(22-y)-(y)=12`

`rarr22-y-y=12`

`rarr22-2y=12`

`rarr22-2y=color(red)(22)=12-color(red)(22)`

`rarr-2y=-10`

`rarr(cancel(-2)y)/color(red)(cancel(-2))=(-10)/color(red)((-2)`

`color(green)(rArry=5`

Substitute the value of `y` to the first equation

`rarrx+y=22`

`rarrx+5=22`

`rarrx=22-5`

`color(green)(rArrx=17`

So, `color(purple)((x,y)=(17,5)`

Hope this helps!!! :)

Answer 2

The two numbers are `17` and `5`.

Explanation:

From the given data we can write:

`x+y=22`
`x-y=12`

From the second equation, we can derive a value for `x`.

`x-y=12`

Add `y` to both sides.

`x=y+12`

In the first equation, replace `x` with `color(red)((y+12))`.

`x+y=22`

`color(red)((y+12))+y=22`

Open the brackets and simplify.

`color(red)(y+12)+y=22`

`2y+12=22`

Subtract `12` from both sides.

`2y=10`

Divide both sides by `2`.

`y=5`

In the first equation, replace `y` with `5`.

`x+y=22`

`x+5=22`

Subtract `5` from both sides.

`x=17`