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I am having difficulty translating this word problem to numbers.
The instructions read: "Translate to a system of equation and solve"

Problem:
The difference between two numbers is 16. Three times the larger number is seven times the smaller. What are the Numbers?

I need to translate the english to math. Then I need to solve it, I’ve tried this and still cannot get it so I ask for help.
Thank you in advance everyone that will answer.

Let’s first figure out what we don’t know, and make those variables.
We need to find 2 numbers that follow those two rules.
Let’s call those 2 numbers x and y.

So the difference between the two numbers is 16.
x – y = 16

Three times the larger number (x) is 7 times the smaller (y).
So, 3 times the larger number is 3 * x = 3x
7 times the smaller = 7 * y = 7y
3x = 7y

One way to solve a linear equation is called "substitution", when you find x = something not related to x, and then plug that into the equation.

So since 3x = 7y,
x = 7y/3

Plug that into x – y = 16

7y/3 – y = 16

Since y = 3y/3 (the threes cancel)

7y/3 – 3y/3 = 16

See the common denominators? Let’s simplify.

4y/3 = 16

Let’s divide both sides by 4

y/3 = 4

Multiply both sides of the equation by 3

y = 12

Now plug again into the equation for y and solve for x

x – y = 16
x – 12 = 16
x = 28

The two numbers are 12 and 28.

Hope this helped!

2 Responses

1. Baganagalagachoogaboogaloogayay! Says:
   December 25th, 2009 at 11:21 pm

   Let’s first figure out what we don’t know, and make those variables.
   We need to find 2 numbers that follow those two rules.
   Let’s call those 2 numbers x and y.

   So the difference between the two numbers is 16.
   x – y = 16

   Three times the larger number (x) is 7 times the smaller (y).
   So, 3 times the larger number is 3 * x = 3x
   7 times the smaller = 7 * y = 7y
   3x = 7y

   One way to solve a linear equation is called "substitution", when you find x = something not related to x, and then plug that into the equation.

   So since 3x = 7y,
   x = 7y/3

   Plug that into x – y = 16

   7y/3 – y = 16

   Since y = 3y/3 (the threes cancel)

   7y/3 – 3y/3 = 16

   See the common denominators? Let’s simplify.

   4y/3 = 16

   Let’s divide both sides by 4

   y/3 = 4

   Multiply both sides of the equation by 3

   y = 12

   Now plug again into the equation for y and solve for x

   x – y = 16
   x – 12 = 16
   x = 28

   The two numbers are 12 and 28.

   Hope this helped!
   References :

2. Kat K Says:
   December 26th, 2009 at 12:11 am

   x – y = 16 and 3(x) = 7(y)

   References :

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