Guess and check table?- Math?

Question

So before I make a guess and check table, I have to figure out this problem. I have to guess two numbers that equal 40 when they're added, and when they're multiplied they have to equal 351. What are the mystery numbers?

Answer 1

This is a typical example of two equations with two variables:

Equation one: X + Y = 40

Equation two: X * Y = 351

How to solve them? The standard method is as follows:

Step 1: Take either of the equations, usually you take the easiest one and rewrite it in the format:

X = "something without X and only Y" or you could use:

Y = "something without Y and only X"

When you've solved about thirty or fourty of these kind of problems you'll know the trick and easily decide between X or Y

Step 2: if you chose: X = "something without X", go to the other equation and wherever you see X, you fill in : "something without X"

Now there are only Y left in the equation, and you should be able to solve it. The result is: Y = "a value"

Step 3: Fill this value of Y in the first equation to find the value of X

Your example:

Equation one: X + Y = 40

Equation two: X * Y = 351

We'll take number one and rewrite it in the format X = "something without X and only Y". That should be easy for you. The result is:

X = 40 - Y

Now fill  this in equation two; wherever you see an X you fill in (40-Y)

X * Y = 351 ==> (40-Y) * Y = 351 ==> 40Y - Y^2 = 351 ==>

Y^2 - 40Y + 351 = 0.

You should have learned already how to solve this:

solution 1: Y = ( 40 +square root(1600-1404) ) / 2 = (40+14) / 2 = 27

solution 2: Y = ( 40 -square root(1600-1404) ) / 2 = (40-14) / 2 = 13

Now you've got two solutions for Y.

Step 3: Fill them in in the first equation to find X:

X = 40 - Y = 40 - 27 = 13

X = 40 - Y = 40 - 13 = 27

So you're two equations have two solutions :

X = 13 and Y = 27 or

X = 27 and Y = 13

If you see these two solutions you might understand why there are two solutions

Answer 2

They are 27 and 13