Question 230·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Which of the following ordered pairs $(x, y)$ satisfies the system below?

\begin{aligned} x + y &= 5 \\[1.4em] x^2 - y &= 7 \end{aligned}

A squared variable alongside a linear equation signals a linear-nonlinear system: a line and a curve may meet more than once. Graph both equations as written in Desmos, then click their intersections to find the ordered pairs that satisfy both. A pair that works in only one equation is not a solution.

Hints

Hint 1
A solution to a system is one ordered pair that makes both equations true. A pair that works in the first equation but misses the squared equation still fails. Where would a solution sit on the two graphs?
Hint 2
The squared variable makes the second graph a parabola, a U-shaped curve. A line can meet it at two points, so check both intersections before matching a point to an answer choice.

Step-by-step

Graph both equations and read the intersections
Step 1
Identify what counts as a solution
The same $(x,y)$ must make $x+y=5$ and $x^2-y=7$ true. The first equation graphs as a line; the squared $x$ makes the second a parabola, a U-shaped curve. They can meet more than once, so a pair on the line alone isn't enough.
Step 2
Find and match the shared points
Type $x+y=5$ and $x^2-y=7$ on separate Desmos lines, then click both crossings. Desmos shows $(-4,9)$ and $(3,2)$ . Each crossing makes both equations true. Don't mix coordinates from different crossings: $(-4,2)$ is not one of these points. A system's solution must satisfy both equations at once. The offered ordered pair that satisfies the system is $(3,2)$ . Choice D.

Question 230·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

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Strategy

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