Difference between revisions of "2018 AMC 10A Problems/Problem 12"
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− | How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following | + | How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations? |
<cmath>x+3y=3</cmath> | <cmath>x+3y=3</cmath> | ||
<cmath>\big||x|-|y|\big|=1</cmath> | <cmath>\big||x|-|y|\big|=1</cmath> |
Revision as of 21:32, 8 February 2018
Contents
Problem
How many ordered pairs of real numbers satisfy the following system of equations?
Solution
The graph looks something like this:
Now it's clear that there are intersection points. (pinetree1)
Solution 2
can be rewritten to . Substituting for in the second equation will give . Splitting this question into casework for the ranges of y will give us the total number of solutions.
Case 1:
will be negative so
Subcase 1:
is positive so and and
Subcase 2:
is negative so . and so there are no solutions ( can't equal to )
Case 2:
Case 3:
will be positive so
Subcase 1:
will be negative so --> . There are no solutions (again, can't equal to )
Subcase 2: y<4/3
will be positive so --> . and Solutions:
NOTE: Please fix this up using latex I have no idea how
Solution by Danny Li JHS
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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 |
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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 12 Problems and Solutions |
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