SOLUTION: Ivan is three years older than his sister Mary. One-third of Ivan's age is two years less than one-half his sister's age one year ago. Find their age's now.

Question 662667: Ivan is three years older than his sister Mary. One-third of Ivan's age is two years less than one-half his sister's age one year ago. Find their age's now.
Found 2 solutions by radh, kevwill:
Answer by radh(108) (Show Source): You can put this solution on YOUR website!
Let's write this algebraically.
Ivan (I) is three years older (+3) than his sister Mary (M). [M+3=I]
One third (1/3) of Ivan's (I) age is 2 years less (-2) than one-half his sister's age (1/2M) one year ago (-1). [1/3I=1/2M-3]
We can now describe this as a system. Let's assume I is X and M is Y. Let's put our equations into x+y=# form.

M+3=I
M=I+3
M-I=3
-X+Y=3

1/3I=1/2M+3
2/3I=M+3
2/3I-M=3
2/3X-Y=3

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

First let . 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 and which are highlighted here:

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 . Remember that the determinant of the 2x2 matrix is . If you need help with calculating the determinant of any two by two matrices, then check out this solver.

Notation note: denotes the determinant of the matrix A.

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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 (since we're replacing the 'x' column so to speak).

Now compute the determinant of to get . Once again, remember that the determinant of the 2x2 matrix is

To find the first solution, simply divide the determinant of by the determinant of to get:

So the first solution is

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

We'll follow the same basic idea to find the other solution. Let's reset by letting 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 (since we're replacing the 'y' column in a way).

Now compute the determinant of to get .

To find the second solution, divide the determinant of by the determinant of to get:

So the second solution is

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Final Answer:

So the solutions are and giving the ordered pair (-18, -15)

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

There's no such thing as a negative age, so just flip that. That means Ivan is 18 and Mary is 15.

Hope this helps!
Answer by kevwill(135) (Show Source): You can put this solution on YOUR website!
I think the original solver got this one wrong.
Let's let i = Ivan's age and m = Mary's age. Ivan is 3 years older than Mary, so

The next one is a little tricky. One third of Ivan's age is two years less than one half of his sister's age one year ago. So

To simplify the rest of the solution, let's multiply both sides of the above equation by 6:

Substituting m+3 for i (from the first equation) gives us

Subtract 2*m from both sides:

Add 15 to both sides:

So Mary is 21 and Ivan is 24.
We can check our answer by inserting our values for i and m into the original second equation:

So our calculations are correct.

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