if the sum of the two numbers is -30 and their difference is -2, find both numbers

Question

if the sum of the two numbers is -30 and their difference is -2, find both numbers

 

Answer 1

We want to begin by rephrasing this question from words to mathematical expressions.  We can see that there are two equations here for two reasons.  The first is that they are telling us there is an addition action (the sum) and the second is there is a subtraction action (the difference).  We also know there are two equations because there are two unknowns and we know it takes "n" equations to find "n" unknowns where "n" just means a number.  Thus one equation for one unknown, two equations for two unknowns, three equations for three unknowns, etc.  Let's proceed by calling our two variables "x" and "y" for convenience noting that we could have called them anything but we chose these because they are common.  Our first statement is that the sum of the two number is -30 so

 

x + y = -30.

 

The second mathematical statement is that their difference is -2 so

 

x - y = -2.

 

Now a little trick I like to use on two equations in two unknowns is called summing the equations.  To do this we put one under the other and either choose to add the two or subtract the two. 

 

x + y = -30

x - y  = -2

 

here we can see that adding the two will eliminate "Y" (make sure you understand why that is so) so we do this operation

 

   x + y = -30

+(x - y = -2)

   2x      = -32

 

Now lets solve that resultant equation for x by dividing by 2

 

x=-32/2=-16

 

and that gives us "x".  We can now substitute "x" back into either of the two equations to find out "y".

 

-16 + y = -30

 

and solving this by adding 16 to both sides leads to 

 

y=-30+16 = -14.

 

So now we have x=-16 and y = -14 but we are not done yet.  We must check to make sure that our answer works by substituting what we found into both equations and making sure they are still true

 

x + y = -16 + (-14) = -30  (remember adding a negative is like subtraction so simplifying)

            -16 - 14      = -30   TRUE!!

 

Now lets do the second equation

 

x - y = -16 - (-14) = -2   (remember that subtracting a negative is like adding)

           -16  + 14   = -2   TRUE!!

 

Because both equations are true are answers is correct and the two numbers are -16 and -14.  If you have problems remembering the adding a negative and subtracting a negative then just remember the multiplying rule that any two signs alike multiplied are positive and any two signs not alike that are multiplied are negative and then just imagine an imaginary coefficient of 1 being multiplied by the factor so that

 

+ (-#) = +1 x (-#)  which must be negative since the signs differ and

- (-#) = -1 x (-#) which must be positive since the signs are alike.

 

Hope all this helps you, not only with your problem, but how to solve others like it.

 

Dave 

 

 

 

Answer 2

If the sum of two numbers is -30 and their difference is -2, find both numbers

 

Identify the unknowns:

Let x = one of the unknown numbers

let y = the other unknown number

 

We have 2 unknowns, so we need two equations relating the unknowns:

Equation 1: the sum of two numbers is -30:  x + y = -30

Equation 2: their difference is -2: x - y = -2

 

We want to eliminate one of the variables so we can solve for the other:

From equation 1:

x + y = -30

So y = -30 - x

 

Substitute 30-x in place of y in equation 2:

x - y = -2

x - (-30 - x) = -2

 

Solve for x

x - (-30 - x) = -2

2x + 30 = -2

2x = -32

x = -16

 

Now solve for y:

y = -30 - x

y = -30 - (-16)

y = -30 + 16

y = -14

 

Check:

x + y = -30

(-16) + (-14) = -30

-30 = -30                     CHECK!

 

x - y = -2

-16 - (-14) = -2

-16 + 14 = -2

-2 = -2                         CHECK!