15Ellen is currently twice as old as Maria, but in6 years, Maria will be 2/3 as old as Ellen. How old isEllen now?16If 2x - 2y = 5 and x + y = 6, what is the valueHow do I do this?

Accepted Solution

Answer:15) Ellen is 12 years old.16) Don't know what value you are looking for but x=(17/4) while y=(7/4) since the set of equations is 2x-2y=5 and x+y=6. Step-by-step explanation:15) Ellen (E) is currently twice as old as Maria (M).Twice means to multiply by 2.E=2M is therefore the equation for this part of the problem.In 6 years, Maria (M) will be 2/3 as old as Ellen (E).  Keep in my Maria as aged 6 years (M+6) and Ellen as aged 6 years (E+6):M+6=(2/3)(E+6).So we have this system to solve:E=2MM+6=(2/3)(E+6).I'm going to replace the E in the second equation with 2M since the first equation says E=2M:M+6=(2/3)(E+6) with E=2M:M+6=(2/3)(2M+6)Distribute:M+6=(4/3)M+(12/3)Reducing 12/3 to 4:M+6=(4/3)M+4Subtract 4 on both sides:M+2=(4/3)MSubtract M on both sides:2=(4/3)M-MFind a common denominator:  2=(4/3)M-(3/3)MCombine like terms:2=(1/3)MMultiply both sides by the reciprocal of 1/3 which is 3:3(2)=3(1/3)M6=1M6=MM=6Maria is six years old.Recall E=2M.So if M=6, then E=2(6)=12.So currently E is twice as old as M since 6(2)=12.In six years, M will be 12 which E will be 18.M (12) is two-thirds as old as E (18) since 12=2/3(18)16)2x-2y=5  x + y=6Your question doesn't state which value it is looking for.Anyways I'm going to solve the system for the point (x,y) and then you can decide which part of this answer to use.I'm going to solve this system by elimination.  Each equation already has the same form: ax+by=c. I just need a column with the variables to be opposite or the same.I'm going to multiply both sides of equation 2 by 2:2x-2y=52x+2y=12Now I have opposites in a column: -2y and 2y.When you add opposites you get 0 so this is what you want to use elimination. We are going to add the equations now:2x-2y=52x+2y=12------------------Adding!4x+0y=174x      =17Divide both sides by 4:  x      =17/4Now if x=17/4 and x+y=6, then we have that (17/4)+y=6.Subtracting 17/4 on both sides gives us y=6-(17/4).Finding a common denominaotr gives us y=(24/4)-(17/4).Simplifying this gives us y=(7/4).The point of intersection (the solution) is (17/4 ,  7/4)