the difference between two number is 11. Twice the smaller plus three times the larger number is 123. what are the numbers?

Question

this is an algerbraic word problem that needs the use of variables.

Answer 1

Let's say a = smaller number and b = larger number. 

Since a is the smaller number, we would set b = a+11. 

Next, as the question states, twice the smaller (2a) plus three times the larger number (3b), gives us 2a + 3b = 123. 

Using the b = a+11, we substitute b as a + 11, so it would give us 2a + 3(a+11) = 123. 

Distribute the 3(a+11) into 3a + 33, and we would get 2a + 3a+33 = 123, simplifying it to 5a = 90. 

Divide both sides by 5 and we would get a = 18, which is the smaller number. 

Since the difference between the 2 number is 11 as the question states, b would be 29. 

Thus, the two numbers are 18 and 29. 

Answer 2

Hello Alexis!

 

You have stumbled upon a problem which we can use a system of equations to solve. There are two unknown numbers here, so let's call them x and y. Without effecting our result, we can let x>y. Now, since "the difference between two numbers is 11", then 

 

(1) x-y=11.

 

Since "twice the smaller plus three times the larger number is 123", then

 

 

(2) 2y+3x=123.

 

Although there are many ways to solve systems of equations, in this case, I suggest using substitution. We will solve (1) for x and replace x in (2) with our new found expression.

 

You're on your own from here Alexis! Feel free to respond if you have further questions.