The sum of two numbers is 57 and the difference is 19. What are the numbers?

Question

What is the largest and smallest number.

Please help.

Answer 1

We can answer this question using a system of equations:

The object is to turn the statements into equations using variables. The first statement, 'The sum of two numbers is 57' is an addition problem which can be represented by the equation:

x + y = 57

'the difference is 19'. This 2nd statement is, of course, referring to the same two numbers, which we have already defined as variables x and y, and is the subtraction of these variables. We can represent this statement with the equation:

x – y = 19

Now, there are a number of different methods that can be used to solve this system of equations. One method that comes to mind is, rewrite the first equation so y is by itself on one side of the equal sign like so:

x + y = 57 ⇒ y = 57 – x

Then, substitute this equation for y into the 2nd equation like so:

x – y = 19 ⇒ x – (57 – x) = 19

After making the substitution, we can see that the new equation is now only in terms of one unknown variable, x, which we can simplify and solve:

x – (57 – x) = 19 ⇒ x – 57 + x = 19 ⇒ 2x – 57 = 19 ⇒ 2x = 76

⇒ x = 38

Lastly, we can now plug in this x-value into either of the two equations and solve for y:

x + y = 57 ⇒ (38) + y = 57 ⇒ y = 57 – 38 ⇒ y = 19

As a final check, we can plug these two values into both equations to make sure they are true:

x + y = 57 and x – y = 19

(38) + (19) = 57 True

(38) – (19) = 19 True

Answer 2

Just like any math problem, this could be solved in a lot of ways.

One way is as follows:

Let x= smaller number

57-x = larger number (Since the sum of the 2 numbers should be 57)

Now, take the difference (larger - smaller)

57-x - x = 19

57-2x=19

-2x=-38 (subtract 57 to both sides)

x=19 (divide both sides by -2)

So, 57-x=57-19=38

Hence, the smaller number is 19, and the larger number is 38.