One number is five less than a second number. Five times the first is 13 more than 3 times the second. Find the numbers.

Answer 1

One number is five less than a second number. Five times the first is 13 more than 3 times the second. Find the numbers.

Based on the prompt given, there's a few hints as to how the equation is created. Less is a key word that suggests subtraction, meaning that the "first number" (X) is 5 more than the second number. Therefore, to make them equal in the equation you need to either set X equal to Y plus 5, or Y equal to X minus 5. This gives us our first equation:

X - 5 = Y OR X = Y + 5

Then, comes the second sentence. Times is a keyword for multiplication, so "5 times the first..." means X times 5 or X * 5. Stating that something "is" usually means equals, giving us X * 5 =. "13 more than 3 times the second," can be a bit tricky so let's break it down. More means addition, and times means multiplication. Therefore, this statement can be equated to "13 + (3 * Y)" or "3Y + 13". This gives us our full second equation:

5X = 3Y + 13

Having both equations, we want to make sure both of them are in the same form so that we can solve them. These two just so happen to line up fairly well:

X = Y + 5

5X = 3Y + 13

Now, let's use the method of elimination. The goal with this method is to elimination either n or s from the equation, so that we can solve for one at a time. To do so, we can de-simplify one of the equations in order to make one of the variables cancel out. Like so:

-3(1X) = -3(1Y + 5) =

-3X = -3Y - 15

Now that we have two equations that'll cancel out, let's add them together to solve for n.

-3X = -3Y - 15

+5X = 3Y + 13

=2X = - 2

Divide both sides by 2... to get X = -1.

Now we can take -1, and plug it into one of our original equations.

1(-1) = Y + 5

-1 = Y + 5

-1 - 5 = -6, therefore Y = -6.

From our calculations, we have deduced that X = -1, and Y = -6. Math can be tricky sometimes, so let's check our work!

5X = 3Y + 13

5(-1) = 3(-6) + 13

-5 = -18 + 13

-5 = -5.

Everything checks out, which means that X = -1, and Y = -6 is correct.

Answer 2

It's easiest to find the answer when writing out what you read. The problem implies that there are two numbers. Numbers which are unidentified can be written as variables.

Here, we can write the first number as x, and the second number as y.

5x = 3y + 13

Now, since we know that the first number is 5 less than the second, we can replace x with y-5:

The equation then turns out to be: 5(y-5) = 3y + 13

Now, we distribute the 5 on the first side, and then add 25 to both sides:

5y - 25 = 3y + 13

5y = 3y + 38

We then subtract 3y from 5y, and divide 38 by the coefficient to get a number:

5y - 3y = 38

2y = 38

y = 19

We now have the second number. Going back to the problem, we know that the first number is 5 less than the second number. So the first number is 14, and the second number is 19.

x = 14, y = 19