Two equations involving the same and , including one with squared terms, form a nonlinear system. Graph both equations in Desmos and find where they meet: that point makes both equations true. Then read the coordinate asked for, not the other one. If you solve by hand, the difference of squares lets you use the given sum without expanding anything.
Hints
- Hint 1
A solution to a system makes both equations true. On a graph, it sits where the two graphs meet. Since both numbers are positive, which part of the graph should you inspect?
- Hint 2
Desmos labels a point as : is the first coordinate, and is the second. Click the meeting point, then check which coordinate the question asks you to report.
Step-by-step
Approach 1: Graph the system in Desmos
Step 1Find where both equations hold
Type and on separate lines. Desmos draws a line and a curve that meet in the upper-right part of the graph. A solution to a system lies on both graphs, and this is the region where and are both positive.
- Step 2
Read the requested coordinate
Click the meeting point. Desmos labels it . In , the first coordinate is , so the requested value is , not . Choice B.
Approach 2: Use the difference of squares
Step 1Factor the squared expression
The squared terms form a difference of squares: . The middle terms cancel when you multiply the factors back out. Factor the equation with :
- Step 2
Use the given sum
One factor is the given sum . Replace that whole factor:
- Step 3
Find the difference
Divide both sides by :
- Step 4
Cancel the terms
Add and so and cancel:
Combine the terms:
- Step 5
Find
Divide both sides by :
So the requested value of is . Choice B.