Question 961·Medium·Math

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A system of equations is shown.

\begin{cases} x + y = 5 \\ xy = 6 \end{cases}

Which of the following is a possible value of $x$ that satisfies the system?

In a nonlinear system, the equations share the same $x$ and $y$ , but a product such as $xy$ makes one graph curved. Enter both equations in Desmos and click their intersections; those shared points satisfy both conditions. Read the first coordinate when the question asks for $x$ , rather than treating the product as an $x$ -value.

Hints

Hint 1
A system requires the same pair $(x,y)$ to make both equations true. Desmos can graph each equation exactly as written, even when one makes a curve. What would a point shared by the two graphs tell you?
Hint 2
An intersection is where two graphs meet, so its coordinates satisfy both equations. Click each crossing and read the labeled points in $(x,y)$ order. Which coordinate gives the value the question asks for?

Step-by-step

Approach 1: Graph both equations

Step 1
Graph the sum equation
Type $x+y=5$ exactly as given. Desmos draws a line; every point on it has coordinates that add to $5$ . You still need the product condition.
Step 2
Graph the product equation
Add $xy=6$ on the next line. Desmos draws a curve with two crossings of the line. At an intersection, where the graphs cross, both equations hold for the same $(x,y)$ . A point on only one graph won't work.
Step 3
Read the possible x-values
Click both crossings. Desmos labels them $(2,3)$ and $(3,2)$ . The first coordinate of $(x,y)$ is $x$ , so $x$ can be $2$ or $3$ . The question asks for a possible value, and $2$ is listed. Choice B.

Approach 2: Match the sum and product mentally

Step 1
Find a pair with the given product
No Desmos needed. The small numbers allow a direct check. A factor pair is two numbers that multiply to a given number. Since $2\cdot3=6$ , the pair $2,3$ meets the product condition.
Step 2
Check the sum
Check the other equation: $2+3=5$ . So $(x,y)=(2,3)$ satisfies both equations, and its $x$ -value is $2$ . You only need one possible value. Choice B.

Question 961·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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