SOLUTION: Tom is half as old as Ellen was three years ago. In 10 years, Ellen will be one-and-one-half times as old as Tom. Would you please show me the formula to use? And the answer

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Question 215999: Tom is half as old as Ellen was three years ago.
In 10 years, Ellen will be one-and-one-half times as old as Tom.
Would you please show me the formula to use? And the answer?
Thank you.
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Tom is half as old as Ellen was three years ago.
In 10 years, Ellen will be one-and-one-half times as old as Tom.
Would you please show me the formula to use? And the answer?
I probably need more info but here is the problem set-up.

Step 1. Let x be Tom's current age and let y be Ellen's age

Step 2. Let x-3 be Tom's age three years ago and y-3 be Ellen's age three years ago

Step 3. Let y-3=2%28x-3%29 be Ellen's age three years ago since Tom is half of Ellen's age three years ago. Simply this equation as follows

y-3=2%28x-3%29

y-3=2x-6

Add 6-y to both sides of the equation to isolate x and y terms on one side and numbers on the right side

y-3%2B6-y=2x-6%2B6-y

3=2x-y

2x-y=3

Step 4. Let y%2B10=3%28x%2B10%29%2F2 be Ellen's age in ten years since Ellen will be one and one half times as old as Tom where 1.5=3%2F2, x+10 is Tom's age in ten years, and y+10 is Ellen's age in ten years.

Multiply by 2 to both sides of the equation to get rid of denominator

2%28y%2B10%29=2%2A3%28x%2B10%29%2F2

2y%2B20=3x%2B30

Subtract 3x-20 to both sides of the equation

2y%2B20-3x-20=3x%2B30-3x-20

-3x%2B2y=10



Step 5. Now we have a linear system of equations from Steps 3 and 4 and is shown below:
2x-y=3
-3x%2B2y=10

Step 6. Now solve the linear system of equations in Step 5 using substitution.

Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
+system%28+%0D%0A++++2%5Cx+%2B+-1%5Cy+=+3%2C%0D%0A++++-3%5Cx+%2B+2%5Cy+=+10+%29%0D%0A++We'll use substitution. After moving -1*y to the right, we get:
2%2Ax+=+3+-+-1%2Ay, or x+=+3%2F2+-+-1%2Ay%2F2. Substitute that
into another equation:
-3%2A%283%2F2+-+-1%2Ay%2F2%29+%2B+2%5Cy+=+10 and simplify: So, we know that y=29. Since x+=+3%2F2+-+-1%2Ay%2F2, x=16.

Answer: system%28+x=16%2C+y=29+%29.



So x=16 and y=29.

Check the ages three ago

x-3=16-3=13 and y-3=29-3=26 where Ellen is twice the age of Tom three years ago...so this is a true statement.

Check the ages ten years from now.

x%2B10=16%2B10=26 and y%2B10=29%2B10=39 or 39=3%2A26%2F2=39 which is another true statement.

Step 7. So Tom is 16 years old and Ellen is 29 years old.

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit
http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit
http://www.FreedomUniversity.TV/courses/Trigonometry.

Good luck in your studies!

Respectfully,
Dr J