Question 96·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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For positive integers $x$ and $y$ that satisfy the system of equations

\begin{cases} x^2 + y = 13,\\ y^2 + x = 19 \end{cases}

what is the value of $x + y$ ?

Your answer

A nonlinear system has two equations sharing unknowns, with a squared variable signaling that the graphs may meet more than once. Use the positive-integer condition to bound the coordinates, then graph both equations in Desmos and click their intersection in that range. The point gives you $x$ and $y$ , but the question asks for their sum.

Hints

Hint 1
A positive integer is a whole number at least $1$ . In the first equation, that makes $x^2$ at most $13-1$ . Which positive integers have squares no larger than that?
Hint 2
Apply the same bound to the second equation. Since $x\ge1$ , $y^2$ is at most $19-1$ . Use that limit to decide how much of the Desmos graph you need to see.
Hint 3
An intersection is a point that makes both equations true. Click the meeting point in the allowed range, then check what the question asks you to do with its two coordinates.

Step-by-step

Bound the integers, then graph the system
Step 1
Limit the possible values of x
Because $y$ is a positive integer, $y\ge1$ . Subtract its smallest possible value from $13$ in the first equation: $x^2\le12$ . Since $4^2=16>12$ , $x$ can only be $1$ , $2$ , or $3$ .
Step 2
Limit the possible values of y
Because $x\ge1$ , subtract its smallest possible value from $19$ in the second equation: $y^2\le18$ . Since $5^2=25>18$ , $y$ can only be $1$ , $2$ , $3$ , or $4$ .
Step 3
Find the point that satisfies both equations
Type both equations as written, one per line. In the graph window shown, click their meeting point. Desmos labels it $(3,4)$ . An intersection lies on both graphs, so it satisfies both equations. It's the only meeting point in the allowed positive-integer range.
Step 4
Calculate the requested sum
The point gives $x=3$ and $y=4$ . Type $3+4$ on a new Desmos line; it prints $7$ . An intersection gives the coordinates, but you must still calculate the expression asked for. So $x+y=7$ . Grid in 7.

Question 96·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