Depth · one concept from first principles · step 2 of 4
Turn the condition into x+y in terms of xy
If , find the value of the expression below.
ExpressionsNumber system
Your answer
How to solve
Strategy Convert to Algebra — The condition only links x+y and xy. So rewrite the condition as x+y in terms of xy, rewrite the big fraction using only x+y and xy, and substitute. Everything should become a multiple of xy, which cancels.
1STEP 1
Rewrite the condition
Add the two fractions over the common denominator xy, then multiply both sides by xy.
1/x+1/y=(x+y)/(xy)=4/3 → x+y=(4/3)xy
The condition is really a statement about x+y compared with xy.
7.EE.A.2Convert To AlgebraThe condition says the same thing as .
Why?
and , because multiplying top and bottom by the same number only renames a fraction.
🧱Scaling top and bottom keeps a fraction's valueMultiplying or dividing a fraction's top and bottom by the same number only renames it — the pieces change size but the amount you hold does not.
Why?
Pieces of the same size add by counting, so the sum 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?
Multiplying both equal sides by keeps them equal and clears the bottom.
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
2STEP 2
Regroup the fraction around x+y
Expand the numerator and collect the x and y terms; the denominator already has 2x+2y=2(x+y).
(x(2-3y)+2y)/(xy-(2x+2y))=(2(x+y)-3xy)/(xy-2(x+y))
Group the pieces so the known combination x+y shows up.
6.EE.A.3Identify Subproblems3STEP 3
Substitute in the numerator
Replace x+y with (4/3)xy.
2×(4/3)xy-3xy=(8/3)xy-(9/3)xy=-(1/3)xy
Now the numerator is just a number times xy.
7.NS.A.2Convert To Algebra4STEP 4
Substitute in the denominator
Do the same below the bar.
xy-2×(4/3)xy=(3/3)xy-(8/3)xy=-(5/3)xy
The denominator is also a number times xy.
7.NS.A.2Convert To Algebra5STEP 5
Cancel xy
Since xy ≠ 0, it cancels, leaving a ratio of two numbers.
(-(1/3)xy)/(-(5/3)xy)=1/3÷5/3=1/5
Dividing two negatives gives a positive.
7.NS.A.2Convert To AlgebraAnswer
1/5
Pick a pair that fits: x=y=3/2 gives 2/3+2/3=4/3. Then the numerator is 3/2×(-5/2)+3=-3/4 and the denominator is 9/4-6=-15/4, and -3/4÷(-15/4)=1/5.
Takeaway
When you can't find x and y, find the combination the question really needs.
- Rewrite the condition
- Regroup the fraction around x+y
- Substitute in the numerator
- Substitute in the denominator
- Cancel xy
Where next?
Another one like thissuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.