three times a number minus another number is negative 3. The sum of the numbers is 11. What are the numbers?

Question

i really need help on this problem

Answer 1

Three times a number minus another number is negative 3. The sum of the numbers is 11. What are the numbers?

 

Let x = one of the numbers

Let y = the other number

 

We have two unknowns, so we need two equations relating the unknowns:

 

Equation 1: Three times a number minus another number is negative 3:  3x - y = -3

Equation 2: The sum of the numbers is 11:  x + y = 11

y = 11-x

 

Let's substitute 11-x in place of y in equation 1 and solve for x:

3x - y = -3

3x - (11 - x) = -3     [Substitute 11-x for y]

4x - 11 = -3             [Combine like terms]

4x = 8                     [Add 11 to both sides]

 

x = 2

 

Now let's solve for y:

y = 11 - x

y = 11 - 2               [x = 2]

 

y = 9

 

Check:

Three times a number minus another number is negative 3:  3(2) - 9 = 6 - 9 = -3   CHECK!

The sum of the numbers is 11:  2 + 9 = 11    CHECK!

 

Answer 2

First, give the numbers in the problem you're looking for names, i.e. create variables for them -- x and y are pedestrian but well respected choices. Then convert the English sentences of the problem into equations using these variables.  So:

 

(1): 3x - y = 3 --and-- (2): x + y = 11

Now, solve one of the equations for one of the variables. A good idea is to choose the equation with the most easily solved for variable. In this case, that would be the (2) and it would just as easy to solve for either x or y:

 

(2a): x = 11 - y --or-- (2b): y = 11 - x

 

You then substitute this variable into the other equation, i.e. (2a) or (2b) into (1).

 

You could solve (1) for x or y, and substituted this x or y into (2).  9 times out of 10 which equation it doesn’t make any difference which equation your start with and which variable you solve for -- you'll wind up doing the same amount of work either way.

 

Substituting (2b) into (1), i.e. (2b → 1):

(3):  3x - (11 - x) = 3

 

Then you solve this equation for the remaining variable, in this case x, i.e. solve (3) for x:

3x - 11 + x = 3 -- distribute the "-" across the terms in the parentheses

(3x + x) - 11 = 3 -- group the variable terms together

4x - 11 = 3 -- combine the terms with the variables

4x + -11 +11 = 3 + 11 -- get the term with the variable by itself on one side of the equation, add 11 to both sides: (-11 + 11 = 0) and (3 + 11 = 14)

4x/4 = 14/4 -- get the variable by itself (divide both sides by 4)

x = 7/2 -- (14/2 = 7 and 2/2 = 1)

 

Now take the variable you just solved for and plug it back into the equation you substituted into the other equation, in this case x into (2b):

 

(x → 2b): y = 11 - (7/2) = 22/2 - 7/2 = 15/2

 

Now, the nice thing about algebra is that you never need a teacher or the answers in the back of the book to know if you got right answer -- you can always substitute your answer back into the equations and see if the value of the answers you've gotten satisfy the equations. So, substitute x and y into (1):

 

(x and y → 1):

3(7/2) - (15/2) = 3 -- 3(7/2) = (3/1)*(7/2) = (3 * 7)/(1 * 2) = 21/2

21/2 - 15/2 = 3 -- 21/2 - 15/2 = (21 - 15)/2

6/2 = 3

3 = 3 -- 3 = 3, so x and y are a solution for equation (1).

 

(x and y → 2):

(7/2) + (15/2) = 11

(7 + 15)/2 = 11

22/2 = 11

11 = 11 -- 11 = 3, so x and y are a solution for equation (2).

 

Because x and y solve both equations, you know you've got the right answer!