2021 Fall AMC 10A Problems/Problem 14
Contents
[hide]Problem
How many ordered pairs of real numbers satisfy the following system of equations?
Solution 1 (Graphing)
The second equation is . We know that the graph of is a very simple diamond shape, so let's see if we can reduce this equation to that form: We now have two separate graphs for this equation and one graph for the first equation, so let's put it on the coordinate plane: We see from the graph that there are intersections, so the answer is .
~KingRavi
Solution 2
From the first equation, we can express in terms of :
which is
The second equation can be rewritten as:
.
This gives us two scenarios to examine:
1.
2.
Case 1:
Substituting ,
.
First, consider the case when . Then,
which is .
Multiplying by 3, we get
which is which is .
However, the discriminant of this quadratic is , which indicates there are no real solutions in this scenario.
Now, we can consider when
which is .
Multiplying by 3, we get
which is .
Now, let , which gives . When we calculate the discriminant, we get . So, the roots are .
Both roots give positive values for , resulting in two values of for each root (one positive and one negative).
Case 2:
Substituting :
.
If , then which is |.
Multiplying by 3, we get
which is which is .
Thus, , leading to or (both giving corresponding values).
If , then which is .
When we multiply through, we get which is .
The discriminant here is .
This gives two more real roots for .
Now,
- Case 1 contributes 2 solutions
- Case 2 contributes 1 solution from and , and 2 solutions from the second sub-case
Thus, counting all solutions gives us a total of 5 unique ordered pairs, and the answer is .
~goofytaipan
Video Solution
~Education, the Study of Everything
Video Solution
https://youtu.be/zq3UPu4nwsE?t=974
Video Solution by WhyMath
~savannahsolver
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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