Question 225·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Consider the system of equations:

\begin{aligned} x + y &= 5 \\[1.4em] x^2 + y^2 &= 13 \end{aligned}

Which ordered pair (x, y) satisfies the system?

A system of equations asks for a pair that satisfies both equations. A sum equation graphs as a line, while a sum of squares graphs as a circle. Type both equations into Desmos and click their intersections. The graphs can meet more than once, so check which ordered pair actually appears among the choices.

Hints

Hint 1
In an ordered pair $(x,y)$ , the first number is $x$ and the second is $y$ . A pair that fails either equation cannot solve the system. Does every choice have coordinates that add to $5$ ?
Hint 2
Desmos can graph the entire circle equation without solving for $y$ . An intersection is a point on both graphs, so click each place where the line and circle meet.

Step-by-step

Graph both equations in Desmos
Step 1
Rule out a pair that fails the sum
In the ordered pair $(2,2)$ , the first number is $x$ and the second is $y$ . Put them into the first equation:
$2+2=4\ne5.$
So $(2,2)$ fails the sum condition. Passing that condition alone wouldn't be enough: a solution must pass the other equation too.
Step 2
Find the points shared by both graphs
Type $x+y=5$ and $x^2+y^2=13$ on separate Desmos lines. Click both intersections, where the line and circle meet. Desmos shows $(2,3)$ and $(3,2)$ . Each intersection satisfies both equations. Only $(2,3)$ appears among the choices, so the requested ordered pair is $(2,3)$ . Choice C.

Question 225·Medium·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