Question

The sum of two numbers is 14 and the diference is 6 what are the two numbers?

Answer 1

`a = 10 and b = 4`.

Explanation:

Let a be the larger and b be the smaller. We know that

`a+b = 14 and a-b = 6`.

Solving the 2nd equation for a,

`a = 6 + b`.

Substituting that expression for a in for the a in the first equation, and then solving for b,

`6+b+b = 14`

`2*b = 14-6 = 8`

`b = 8/2 = 4`
Therefore `a = 6+4 = 10`.

I hope this helps,
Steve

Answer 2

`10` and `4`.

Explanation:

Let's call the two numbers `x` and `y`. We know that `x+y=14` and `x-y=6`. We could guess and check, but there is an easier way. We can use elimination to solve for `x` and `y`, using *just these two equations! *

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Why?
We can rewrite these two equations as `1x+1y=14` and `1x+(-1y)=6`. Note that `x`'s coefficients are the same, so we can subtract the equations and `x` will disappear, leaving us with the solution for `y`. Also note that `y`'s coefficients are additive inverses, so we can add the equations and `y` will disappear, giving us the solution for `x`! That's both variables! Here we go:

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Subtracting (solution for `y`):
`cancel(x)+y=14`
`cancel(x)-y=6`
#y-(-y)=8 #y+y=8# #2y=8# #y=4# **So, one of the numbers is #4#.**

Adding (solution for `x`)
`xcancel(+y)=14`
`xcancel(-y)=6`
`2x=20`
`x=10`
So, the other number is `10`.

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Let's check:
`x+y=14`
`10+4=14`
`14=14`

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`x-y=6`
`10-4=6`
`6=6`
We're good!

So, the two numbers are `10` and `4`.