Question 1054·Hard·Math

0 / 2000

Positive real numbers $a$ and $b$ satisfy

\begin{cases} a^{2}+b=7,\\[4pt] b^{2}+a=11. \end{cases}

What is the value of $a^{2}+b^{2}$ ?

Your answer

Two equations with squared variables form a nonlinear system: a solution must make both equations true. Graph both equations in Desmos, using $x$ for the first number and $y$ for the second. Because the numbers must be positive, use a crossing where both coordinates are positive. Read the requested expression, not just the point or the sum of the given totals.

Hints

Hint 1
A system needs the same pair of numbers to satisfy both equations. To graph it, replace $a$ with $x$ and $b$ with $y$ in each equation.
Hint 2
The positive quadrant is the part of the graph where both coordinates are greater than $0$ . A crossing there gives a pair that satisfies both equations. Which coordinate represents $a$ ?
Hint 3
A sum of squares means square each number separately, then add. Adding the two totals in the given equations also includes terms that aren't squared.

Step-by-step

Graph the system in Desmos
Step 1
Graph both equations
Desmos uses $x$ and $y$ for its graph axes, so let $x=a$ and $y=b$ . Type $x^{2}+y=7$ and $y^{2}+x=11$ , keeping the same replacements in both equations. Each graph shows pairs that satisfy one equation; where they cross, a pair satisfies both.
Step 2
Read the positive solution
Click the crossing where both coordinates are positive, as the question requires. Desmos labels it $(2,3)$ . Since $x=a$ and $y=b$ , this gives $a=2$ and $b=3$ .
Step 3
Find the sum of the squares
The question asks you to square each number before adding, not to add the two totals from the equations. Type $2^{2}+3^{2}$ on a new Desmos line; it displays $13$ . So $a^{2}+b^{2}=13$ . Grid in 13.

Question 1054·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

Hints

Step-by-step

Lessons that teach this