How to calculate two integers x and y when the sum of x+y is known and difference between x and y is known

Question

I know this has a simple solution, but I can not come up with it.

As an example, if two sports teams play a game and Team X beats Team Y by 28 points and the sum of Team X's score and Team Y's score is 62, then how many points did Team X score and how many points did Team Y score?

I am interested in the formula or way to set up this problem.

Thank you.

This is just a linear simultaneous equation. You are given $(x+y)$ and $(x-y)$ so you can just find for example $x=\frac12((x+y)+(x-y))$ — Peter Foreman, Sep 11 '19 at 15:02
If you know $x+y$ and you know $x-y$, then by adding these two together... you get $(x+y)+(x-y)=2x$ and this is enough information to find $x$. Continue — JMoravitz, Sep 11 '19 at 15:03
For the more general problem... this is a system of linear equations. The wikipedia article covers several of the common methods for solving such a system. — JMoravitz, Sep 11 '19 at 15:04

score 0 · Answer 1 · answered Sep 11 '19 at 15:03

0

$$X-Y=28\tag1$$

$$X+Y=62\tag2$$

add $(1) $ and $(2)$: $$2X=28+62$$

subtract $(1)$ from $(2)$:

$$2Y=62-28$$

Can you take it from here?

answered Sep 11 '19 at 15:03

J. W. Tanner

60,406

Another way to think of it: the average score is $62/2=31$; the difference between each score and the average is $28/2=14$ – J. W. Tanner Sep 11 '19 at 15:11

score 0 · Answer 2 · answered Sep 11 '19 at 15:18

Let the sum be $S$, and the difference be $D$;

We have

$$x+y=S$$ and $$x-y=D$$

Adding the two equations, we get:

$$2x=S+D \implies \boxed{x=\frac{S+D}{2}}$$

Substituting the value of $x$ obtained earlier in any one of the two equations above, we get:

$$\frac{S+D}{2}+y=S$$

Solving for $y$, we get:

$$y=S-\frac{S+D}{2} \implies \boxed{y=\frac{S-D}{2}}$$

In this case, we have $S=62$ and $D=28$. Therefore,

$$x=\frac{62+28}{2}=\frac{90}{2}=45$$

and

$$y=\frac{62-28}{2}=\frac{34}{2}=17$$

How to calculate two integers x and y when the sum of x+y is known and difference between x and y is known

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