Question 988·Medium·Math

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The system of equations is shown below:

2x + y = 10 \\[1.4em] xy = 12

If $(x, y)$ is a solution to the system above, what is the smaller possible value of $x$ ?

Your answer

An intersection of two graphs is a point that satisfies both equations. A product like $xy$ makes one graph curved, so it can cross a line more than once. Type both equations as written in Desmos, click every crossing, and compare the coordinates the question asks for. Stopping at the first crossing can miss the smaller value.

Hints

Hint 1
A solution to a system is a point that satisfies both equations. On a graph, that means an intersection. Because the $xy$ term makes one graph curved, could there be more than one intersection?
Hint 2
In an ordered pair $(x,y)$ , the first coordinate is $x$ . Once you’ve clicked every crossing, compare the first numbers, not the second numbers.

Step-by-step

Read both intersections in Desmos
Step 1
Graph both equations
Type $2x+y=10$ and $xy=12$ on separate lines in Desmos. The $xy$ term makes the second graph a curve, so look for every crossing with the line. The graphs cross twice in the shown window; each crossing satisfies both equations.
Step 2
Compare the x-coordinates
Click both crossings. Desmos labels them $(2,6)$ and $(3,4)$ . In $(x,y)$ , $x$ is the first coordinate, so the possible $x$ -values are $2$ and $3$ . Compare the requested coordinate at every intersection. The smaller possible value of $x$ is $2$ . Grid in 2.

Question 988·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this