Depth · one concept from first principles · step 2 of 4
Rename repeated reciprocals to get a linear system
Solve the equation .
Expressions
Your answer
How to solve
Strategy Convert to Algebra — Reorganize the chain A=B=C into two equations that each equal 1. Only two building blocks appear, 1/(x+y) and 1/(x-y), so giving each a new name turns the problem into an ordinary linear system. Then break the rest into two small subproblems: undo the renaming, and solve for x and y.
1STEP 1
Split the chain into two equations
Each of the first two expressions equals the last one, 1.
5/(x+y)-2/(x-y)=1, 1/(x-y)-3/(x+y)=1
Pairing each part with the plain number 1 keeps the equations simple.
8.EE.C.8Organize Information In More Ways2STEP 2
Rename the repeated reciprocals
Let A=1/(x+y) and B=1/(x-y). Both equations become linear in A and B.
5A - 2B = 1, -3A + B = 1
The fractions were never the real difficulty; they are just two quantities used twice.
7.EE.A.2Convert To AlgebraWith and , the two equations become and .
Why?
is five pieces each the size of , so it is ; in the same way is and is .
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
Why?
names the same number everywhere it is used, so swapping each copy for does not change the value of either side.
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
3STEP 3
Solve for A and B
From the second equation B=1+3A. Substitute into the first equation.
5A - 2(1+3A) = 1 → -A - 2 = 1 → A = -3, B = 1 + 3 × (-3) = -8
This is now a routine two-variable linear system.
8.EE.C.8Convert To Algebra4STEP 4
Undo the renaming
If 1/(x+y)=-3 then x+y=-1/3, and if 1/(x-y)=-8 then x-y=-1/8.
x + y = -1/3, x - y = -1/8
Taking reciprocals of both sides is safe because neither side is 0.
8.EE.C.8Identify Subproblems5STEP 5
Add and subtract to get x and y
Adding the two equations gives 2x; subtracting them gives 2y.
2x = -8/24 - 3/24 = -11/24 → x = -11/48, 2y = -8/24 + 3/24 = -5/24 → y = -5/48
A sum and a difference always pin down the two numbers.
8.EE.C.8Identify SubproblemsAnswer
x=-11/48, y=-5/48
Now x+y=-16/48=-1/3 and x-y=-6/48=-1/8. Then 5/(x+y)-2/(x-y)=-15+16=1 and 1/(x-y)-3/(x+y)=-8+9=1, so every part of the chain equals 1.
Takeaway
When the same messy piece keeps showing up, give it a new name and the problem often becomes one you already know how to solve.
- Split the chain into two equations
- Rename the repeated reciprocals
- Solve for A and B
- Undo the renaming
- Add and subtract to get x and y
Where next?
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