Question 595521
Ten years from now Charlotte will be twice as old as Emily will be then. Five years ago Charlotte was 5 times as old as Emily was then. How old are both now?
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Hi, there!
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[I] Define the variables.
C = Charlotte's age now
E = Emily's age now
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[II] Write a system of equations.
In the first sentence, we translate "ten years from now Charlotte will be" into algebra as C+10.
We translate "as old as Emily will be then" into algebra as E+10.
"Charlotte will be twice as old as Emily" so our first equation is
{{{C+10=(2)(E+10)}}}
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In the second sentence, we translate "Five years ago Charlotte was" as C-5.
We translate "Emily was then" as E-5.
"Charlotte was five times as old as Emily" so our second equation is
{{{C-5=(5)(E-5)}}}
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[III] Simplify and solve the system of equations.
{{{C+10=(2)(E+10)}}}
{{{C+10=2E+20)}}}
{{{C=2E+10)}}}
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Now that we have isolate the variable C, we can substitute 2E+10 for C in the second equation. Simplify and solve for E.
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{{{C-5=(5)(E-5)}}}
{{{(2E+10)-5=(5)(E-5)}}}
{{{2E+5=5E-25}}}
{{{-3E+5=-25}}}
{{{-3E=-30}}}
{{{E=10}}}
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Now we know that Emily is 10 years old. To find Charlotte's age, substitute 10 for E in either equation.
{{{C+10=(2)(E+10)}}}
{{{C+10=(2)(10+10)}}}
{{{C+10=(2)(20)}}}
{{{C+10=40}}}
{{{C=30}}}
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We see that Charlotte is 30 years old. Now go back to the original sentences and make sure that the ages work.
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Ten years from now Charlotte will be twice as old as Emily will be then. 
[30+10] is twice as old as [10+10], or 40 is twice as old as 20. YES!
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Five years ago Charlotte was 5 times as old as Emily was then.
[30-5] was 5 times as old as [10-5], or 25 is five times 5. YES!
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I hope that this helps! Feel free to email if you have questions.
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Ms.Figgy
math.in.the.vortex@gmail.com