SOLUTION: Ten years ago a father was seven times as old as his son, two years hence twice his age will be equal to five times his son's age. What are their present ages?

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Question 190313: Ten years ago a father was seven times as old as his son, two years hence twice
his age will be equal to five times his son's age. What are their present ages?
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Let the current ages be "x" and "y" (x for father, y for son)
Given
%28x-10%29+=+7%28y-10%29
x-10+=+7y+-+70
x+-+7y+=+-60
and
2%28x%2B2%29+=+5%28y%2B2%29
2x+%2B+4+=+5y+%2B+10
2x+-+5y+=+6
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax-7%2Ay=-60
2%2Ax-5%2Ay=6

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-7%2Ay=-60-1%2AxSubtract 1%2Ax from both sides

y=%28-60-1%2Ax%29%2F-7 Divide both sides by -7.


Which breaks down and reduces to



y=60%2F7%2B%281%2F7%29%2Ax Now we've fully isolated y

Since y equals 60%2F7%2B%281%2F7%29%2Ax we can substitute the expression 60%2F7%2B%281%2F7%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B-5%2Ahighlight%28%2860%2F7%2B%281%2F7%29%2Ax%29%29=6 Replace y with 60%2F7%2B%281%2F7%29%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax-5%2A%2860%2F7%29-5%281%2F7%29x=6 Distribute -5 to 60%2F7%2B%281%2F7%29%2Ax

2%2Ax-300%2F7-%285%2F7%29%2Ax=6 Multiply



2%2Ax-300%2F7-%285%2F7%29%2Ax=6 Reduce any fractions

2%2Ax-%285%2F7%29%2Ax=6%2B300%2F7Add 300%2F7 to both sides


2%2Ax-%285%2F7%29%2Ax=42%2F7%2B300%2F7 Make 6 into a fraction with a denominator of 7


2%2Ax-%285%2F7%29%2Ax=342%2F7 Combine the terms on the right side



%2814%2F7%29%2Ax-%285%2F7%29x=342%2F7 Make 2 into a fraction with a denominator of 7

%289%2F7%29%2Ax=342%2F7 Now combine the terms on the left side.


cross%28%287%2F9%29%289%2F7%29%29x=%28342%2F7%29%287%2F9%29 Multiply both sides by 7%2F9. This will cancel out 9%2F7 and isolate x

So when we multiply 342%2F7 and 7%2F9 (and simplify) we get



x=38 <---------------------------------One answer

Now that we know that x=38, lets substitute that in for x to solve for y

2%2838%29-5%2Ay=6 Plug in x=38 into the 2nd equation

76-5%2Ay=6 Multiply

-5%2Ay=6-76Subtract 76 from both sides

-5%2Ay=-70 Combine the terms on the right side

cross%28%281%2F-5%29%28-5%29%29%2Ay=%28-70%2F1%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5 on the left side.

y=-70%2F-5 Multiply the terms on the right side


y=14 Reduce


So this is the other answer


y=14<---------------------------------Other answer


So our solution is

x=38 and y=14

which can also look like

(38,14)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax-7%2Ay=-60
2%2Ax-5%2Ay=6

we get


graph of 1%2Ax-7%2Ay=-60 (red) and 2%2Ax-5%2Ay=6 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (38,14). This verifies our answer.


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Check:

Plug in (38,14) into the system of equations


Let x=38 and y=14. Now plug those values into the equation 1%2Ax-7%2Ay=-60

1%2A%2838%29-7%2A%2814%29=-60 Plug in x=38 and y=14


38-98=-60 Multiply


-60=-60 Add


-60=-60 Reduce. Since this equation is true the solution works.


So the solution (38,14) satisfies 1%2Ax-7%2Ay=-60



Let x=38 and y=14. Now plug those values into the equation 2%2Ax-5%2Ay=6

2%2A%2838%29-5%2A%2814%29=6 Plug in x=38 and y=14


76-70=6 Multiply


6=6 Add


6=6 Reduce. Since this equation is true the solution works.


So the solution (38,14) satisfies 2%2Ax-5%2Ay=6


Since the solution (38,14) satisfies the system of equations


1%2Ax-7%2Ay=-60
2%2Ax-5%2Ay=6


this verifies our answer.