Question 99·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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The ordered pair $(x, y)$ satisfies the system of equations

\begin{cases} x^2 - y = 1 \\ y^2 - x = 7 \end{cases}

The system has exactly two real solutions. What is the greater of the two possible values of $x + y$ ?

Your answer

A nonlinear system contains squared variables, so its graphs may meet more than once. Graph both equations as written in Desmos and click every intersection; each crossing gives a pair that satisfies both equations. Compare the requested expression across all solution pairs, not just one coordinate. The rightmost point won't always give the greatest value of an expression like $x+y$ .

Hints

Hint 1
An intersection is a point on both graphs, so it satisfies both equations. Desmos can graph the equations as written; you don't need to rearrange either one. Where do the graphs meet?
Hint 2
A larger $x$ alone doesn't always mean a larger $x+y$ . Compare both coordinates of the two solution points. Does one point have both a larger $x$ and a larger $y$ ?

Step-by-step

Graph the system and compare the sums
Step 1
Find both solution points
Type both equations as written on separate lines in Desmos. Each intersection, where the graphs meet, satisfies both equations. Click the two intersections: Desmos shows approximately $(-1.811,2.278)$ and $(2,3)$ .
Step 2
Choose the point with the greater sum
At $(2,3)$ , both coordinates are larger than at $(-1.811,2.278)$ : $2>-1.811$ and $3>2.278$ . So $(2,3)$ must also have the greater $x+y$ . Don't stop at the larger $x$ or $y$ alone; the question asks for their sum.
Step 3
Add the coordinates
Type $2+3$ in Desmos. It prints $5$ , the greater possible value of $x+y$ . Grid in 5.

Question 99·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this