SOLUTION: A first number plus twice a second number is 13. Twice the first number plus the second totals 32. Find the numbers

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Question 1199298: A first number plus twice a second number is 13. Twice the first number plus the second totals 32. Find the numbers
Found 3 solutions by ikleyn, MathLover1, greenestamps:
Answer by ikleyn(52786) About Me  (Show Source):
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.
A first number plus twice a second number is 13.
Twice the first number plus the second totals 32.
Find the numbers
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    x + 2y = 13     (1)

   2x +  y = 32     (2)


From equation (1),  express x = 13-2y  and substitute it into equation (2).  You will get

    2*(13-2y) + y = 32.


Simplify and find y

    26 - 4y + y = 32

    26 - 32 = 4y- y

       -6   =   3y

        y   =   -6/3 = -2.


Then x = 13 - 2y = 13 - 2*(-2) = 13 + 4= 17.


ANSWER.  1st number is 17.  2nd number is -2.

Solved.



Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

let first number be x and second number y
if a first number plus twice a second number is 13, we have

x%2B2y=13 ..........solve for x
x=13-2y ........eq.1

if twice the first number plus the second totals 32, we have
2x%2By=32..........eq.2, substitute x from eq.1
2%2813-2y%29%2By=32
26-4y%2By=32
26-3y=32
26-32=3y
-6=3y
y=-2

go to

x=13-2y ........eq.1, substitute y
x=13-2%28-2%29
x=13%2B4
x=17

the numbers are:17 ,-2




Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


You have received two good responses showing typical algebraic solutions. For this particular problem, because of the form of the given information, there is a different path to the solution that conceptually is much easier, although the formal algebra for this path is not more efficient.

A first number plus twice a second number is 13. Twice the first number plus the second totals 32.

Informally, using only mental arithmetic and logical reasoning, we can solve the problem like this:

(1) Add the two pieces of given information to find that 3 times the first number plus 3 times the second number is 45.
(2) From that, conclude that the sum of the two numbers is 15.
(3) Compare (2) to the fact that the first number plus twice the second number is 13 to conclude that the second number is -2.
(4) Use (2) and (3) to conclude that the first number is 17.

ANSWERS: 17 and -2

While that method of solving the problem informally is fast and easy, using formal algebra for that path to the solution is not very efficient....

(1) x%2B2y=13
(2) 2x%2By=32
(3) 3x%2B3y=45 [add (1) and (2)]
(4) x%2By=15 [divide by 3]
(5) y=-2 [subtract (4) from (1)]
(6) x=17 [from (4) and (5)]