SOLUTION: Ten years from now, A will be twice as old as B. Five years ago, A was three times as old as B. What are the present ages of A and B?

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Question 611044: Ten years from now, A will be twice as old as B. Five years ago, A was three times as old as B. What are the present ages of A and B?
Found 2 solutions by radh, Earlsdon:
Answer by radh(108) About Me  (Show Source):
You can put this solution on YOUR website!
Let's map this out mathematically
Then years from now, (+10) A will be twice as old as B (2b). Five years ago, (-5) A was three times as old as B (3b).

So, that means:
2b%2B10=a
and
3b-5=a.

Let's simplify that and move both variables on one side of the equation to get:
2b-a=-10
and
3b-a=5

We're going to substitute this into the solver. x=b and y=a.

Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%282%2Ax%2B-1%2Ay=-10%2C3%2Ax%2B-1%2Ay=5%29



First let A=%28matrix%282%2C2%2C2%2C-1%2C3%2C-1%29%29. This is the matrix formed by the coefficients of the given system of equations.


Take note that the right hand values of the system are -10 and 5 which are highlighted here:
system%282%2Ax%2B-1%2Ay=highlight%28-10%29%2C3%2Ax%2B-1%2Ay=highlight%285%29%29



These values are important as they will be used to replace the columns of the matrix A.




Now let's calculate the the determinant of the matrix A to get abs%28A%29=%282%29%28-1%29-%28-1%29%283%29=1. Remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc. If you need help with calculating the determinant of any two by two matrices, then check out this solver.



Notation note: abs%28A%29 denotes the determinant of the matrix A.



---------------------------------------------------------



Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bx%5D (since we're replacing the 'x' column so to speak).


A%5Bx%5D=%28matrix%282%2C2%2Chighlight%28-10%29%2C-1%2Chighlight%285%29%2C-1%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%28-10%29%28-1%29-%28-1%29%285%29=15. Once again, remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc



To find the first solution, simply divide the determinant of A%5Bx%5D by the determinant of A to get: x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%2815%29%2F%281%29=15



So the first solution is x=15




---------------------------------------------------------


We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C2%2C-1%2C3%2C-1%29%29 again (this is the coefficient matrix).




Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5By%5D (since we're replacing the 'y' column in a way).


A%5Bx%5D=%28matrix%282%2C2%2C2%2Chighlight%28-10%29%2C3%2Chighlight%285%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%282%29%285%29-%28-10%29%283%29=40.



To find the second solution, divide the determinant of A%5By%5D by the determinant of A to get: y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%2840%29%2F%281%29=40



So the second solution is y=40




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Final Answer:




So the solutions are x=15 and y=40 giving the ordered pair (15, 40)




Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.




So, person A is 40 and person B is 15. :)


Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
You can express these two situations algebraically:
A+10 = 2(B+10) "Ten years from now (A+10), A will be twice as old as B (will be ten years from now) 2(B+10)" and
A-5 = 3(B-5) "Five years ago (A-5), A was three times as old as B (was five years ago), (3(B-5))" Solve this for A to get:
A = 3(B-5)+5 and simplify to get:
A = 3B-15+5 or
A = 3B-10 Now substitute into the first equation:
(3B-10)+10 = 2(B+10) Simplify and solve for B.
3B =2B+20
highlight%28B+=+20%29 and...
A = 3B-10
A = 3(20)-10
A = 60-10
highlight%28A+=+50%29
Check:
A+10 = 2(B+10)
50+10 = 2(20+10)
60 = 2(30)
60 = 60 and...
A-5 = 3(B-5)
50-5 = 3(20-5)
45 = 3(15)
45 = 45