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ratelena [41]
3 years ago
15

How many solutions does the equation have, x1 + x2 + x3 = 10 , where x1 , x2, and x3 are non-negative integers?

Mathematics
2 answers:
MAVERICK [17]3 years ago
8 0

Non-negative integers are positive integers or zero.

1. When x_1=0, then there are such possible cases for x_2 and x_3:

  • x_2=0,\ x_3=10;
  • x_2=1,\ x_3=9;
  • x_2=2,\ x_3=8;
  • x_2=3,\ x_3=7;
  • x_2=4,\ x_3=6;
  • x_2=5,\ x_3=5;
  • x_2=6,\ x_3=4;
  • x_2=7,\ x_3=3;
  • x_2=8,\ x_3=2;
  • x_2=9,\ x_3=1;
  • x_2=10,\ x_3=0.

In total 11 solutions for x_1=0.

2. For x_1=1, there are such possible cases for x_2 and x_3:

  • x_2=0,\ x_3=9;
  • x_2=1,\ x_3=8;
  • x_2=2,\ x_3=7;
  • x_2=3,\ x_3=6;
  • x_2=4,\ x_3=5;
  • x_2=5,\ x_3=4;
  • x_2=6,\ x_3=3;
  • x_2=7,\ x_3=2;
  • x_2=8,\ x_3=1;
  • x_2=9,\ x_3=0.

In total 10 solutions for x_1=1.

3. This process gives you

  • for x_1=2 - 9 solutions;
  • for x_1=3 - 8 solutions;
  • for x_1=4 - 7 solutions;
  • for x_1=5 - 6 solutions;
  • for x_1=6 - 5 solutions;
  • for x_1=7 - 4 solutions;
  • for x_1=8 - 3 solutions;
  • for x_1=9 - 2 solutions;
  • for x_1=10 - 1 solution.

4. Add all numbers of solutions:

11+10+9+8+7+6+5+4+3+2+1=66.

Answer: there are 66 possible solutions (with non-negative integer variables)

Musya8 [376]3 years ago
3 0

Answer:

There are total 66 solutions of the equations.

Step-by-step explanation:

We can find the solution by fixing the first value i.e. x_1 at a time and shifting the other two values and keep on doing for all the possible values of x_1

Hence, the possible cases are as follows:

x_1-x_2-x_3

0-10-0      1-9-0      2-8-0      3-0-7     4-0-6    5-0-5      6-0-4    

0-0-10      1-0-9      2-0-8      3-7-0     4-6-0     5-5-0     6-4-0

0-1-9         1-1-8       2-1-7       3-1-6      4-1-5      5-1-4       6-1-3

0-9-1         1-8-1       2-7-1       3-6-1      4-5-1      5-4-1       6-3-1

0-2-8        1-2-7      2-3-5      3-2-5      4-2-4     5-2-3      6-2-2

0-8-2        1-7-2      2-4-3      3-5-2      4-4-2     5-3-2

0-3-7         1-3-6      2-3-4      3-3-4      4-3-3

0-7-3         1-6-3      2-2-6      3-4-3

0-4-6         1-4-5      2-6-2

0-6-4         1-5-4

0-5-5

------------------------------------------------------------------------------------------------------------

7-0-3        8-0-2       9-0-1     10-0-0

7-3-0        8-2-0       9-1-0

7-1-2          8-1-1

7-2-1

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