Depth · one concept from first principles · step 2 of 4
Emma and Noah each solved the system of equations , . Emma misread the constant and got , . Noah misread the constant and got , . Find the solution of the original system.
Emma's mistake was in a, which appears only in the first equation. Her pair x=3, y=-3 therefore satisfies the second equation, 2x-by=3.
A wrong answer is still a correct solution of the equation the student did not change.
8.EE.C.8Change Focus Count The ComplementEmma's pair , satisfies the correctly copied equation , and that fixes .
Emma's wrong appears only in the first equation, so her second equation was the true , and a solution of her system makes every one of her equations come out equal, that one included.
Putting , into gives , which settles .
times is , because multiplying two numbers with like signs gives a positive.
Taking off both sides and then dividing both sides by keeps the equation balanced, leaving .
Noah misread b, so the first equation ax+y=-1 was copied correctly, and x=1, y=2 satisfies it.
The same idea, used on the other equation, gives the other constant.
8.EE.C.8Change Focus Count The ComplementWith a=-3 and b=-1 the system is -3x+y=-1 and 2x+y=3. Subtracting the first from the second removes y.
Both equations have +y, so subtraction cancels it at once.
8.EE.C.8Convert To AlgebraSubstitute x=4/5 into 2x+y=3.
Fractions are fine here; nothing says the solution must be whole numbers.
8.EE.C.8Convert To AlgebraA misread constant only spoils its own equation, so a wrong answer still tells you something true about the other one.
A parent dashboard for the family lives at sensimlab.com.
The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Multiplicative Ratio with Integrality Constraints. Why the shapes repeat →