Questions on 3 simultaneous equations (3D vector planes)

In summary, by solving the given systems using elimination, we can determine the values of "a" or "A" and "B" that result in a unique solution, infinite solutions, or no solution. For the first two questions, "a" can be any number except for the value that makes any of the denominators equal to 0. For the third question, "A" can be a specific value that makes the denominator equal to 0, while "B" can be any number except for the value that makes the numerator equal to 0.
  • #1
Charismaztex
45
0

Homework Statement



1) Determine, by elimination, value of a (if any) such that the given system will have a unique solution

[itex]x+2y+3z=2,

2x+2y+az=0,

3x+2y+z=0[/itex]

2) Determine, by elimination, values of a (if any) such that the given system:

a) Is consistent with and infinity of solutions;
b) has a unique solution;
c) is inconsistent, with no solution

[itex]x+2y+3z=a,

x+y+z=0,

3x+2y+z=0[/itex]

3) consider the following system of 3 equations in x,y, and z

[itex] 2x+2y+2z=9,

x+3y+4z=5,

Ax+5y+6z=B[/itex]

Give possibly values of A and B in the third equation which make this system:

a) inconsistent
b) consistent but with an infinite number of solutions

Homework Equations



N/A

The Attempt at a Solution



1) By a unique solution I'm presuming that all three planes meet at the same point. Would this be to solve and get a so that x,y, and z have a unique value?

2) Consistent with infinity of solutions, is that when we get a situation 0=0? Possibly with at least 2 of the plane equations the same, or intersecting like the "spine" of a book. So a value of a to get some sort of 0=0?

would inconsistent be a situation when we get 0=a number (a nonsense statement)?

To be honest, I have no idea how to approach the question which as to determine values of a or A and B. What sort of method would be suitable?

Thanks in advance,
Charismaztex
 
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  • #2
Charismaztex said:

Homework Statement



1) Determine, by elimination, value of a (if any) such that the given system will have a unique solution

[itex]x+2y+3z=2,

2x+2y+az=0,

3x+2y+z=0[/itex]

2) Determine, by elimination, values of a (if any) such that the given system:

a) Is consistent with and infinity of solutions;
b) has a unique solution;
c) is inconsistent, with no solution

[itex]x+2y+3z=a,

x+y+z=0,

3x+2y+z=0[/itex]

3) consider the following system of 3 equations in x,y, and z

[itex] 2x+2y+2z=9,

x+3y+4z=5,

Ax+5y+6z=B[/itex]

Give possibly values of A and B in the third equation which make this system:

a) inconsistent
b) consistent but with an infinite number of solutions

Homework Equations



N/A

The Attempt at a Solution



1) By a unique solution I'm presuming that all three planes meet at the same point. Would this be to solve and get a so that x,y, and z have a unique value?
Yes, just go ahead and solve the equations with the "a" in there. At some point, perhaps more than once, you will have to divide by an expression involving a. The system has a unique solution if, each time you have to divide by an expression involving a, that expression is NOT 0. a can be any number that does not make any of those expressions equal to 0.

2) Consistent with infinity of solutions, is that when we get a situation 0=0? Possibly with at least 2 of the plane equations the same, or intersecting like the "spine" of a book. So a value of a to get some sort of 0=0?
Yes. Here, a will be such that, at some point, you have to "divide by 0" but the number you are dividing into is also 0.

would inconsistent be a situation when we get 0=a number (a nonsense statement)?
Right. Again, a must be such that it makes some expression you divide by 0 but now the value you are dividing into is NOT 0.

To be honest, I have no idea how to approach the question which as to determine values of a or A and B. What sort of method would be suitable?

Thanks in advance,
Charismaztex
The best way to do this would be to actually go ahead and solve for x, y, and z, equal to fractions with a in numerator and denominator. There will be a unique solution if a is such that none of those denominators is 0. There will be an infinite number of solutions if a is such that at least one of the denominators is 0 but so are the corresponding numerators. There will be no solution if a is such that there exist at least one denominator equal to 0 and the corresponding numerator is not equal to 0.
 
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  • #3
Thanks for the reply. Using your method, I've successfully found for Q1) that a is anything except 2 (confirmed by inputting any number except 2 to get a unique solution on the calculator).

For Q2) I've found that no matter how I manipulate the equations, it always comes out to be something like, for example, [itex]y+2z=a, y+2z=0 [/itex]. Hence I can see that a must be 0 for infinite solutions, and a is any number other than 0 to be inconsistent with no solutions. I am stymied, however, at part b) where "a" should be a number where there is a unique solution. I can only see that a is either consistent with a=0 or inconsistent with a=any number, with not third option.

For Q3) I've worked out A=3 (in my equation, I got A=3 to make the denominator 0) and B=14 (I made the numerator =0 and solved for B) for infinite solutions and B=any number except 14 for inconsistent solution. I am wondering if this is the correct way to approach this.

Thanks,
Charismaztex
 

1. What are simultaneous equations?

Simultaneous equations are a set of equations with multiple variables that are solved simultaneously to find the values of the variables that satisfy all the equations.

2. What is the significance of 3D vector planes in simultaneous equations?

In 3D vector planes, the variables are represented as vectors and the equations are represented as planes in 3D space. This allows for a geometric interpretation of the solutions to the simultaneous equations.

3. How do you solve 3 simultaneous equations?

There are multiple methods to solve 3 simultaneous equations, such as substitution, elimination, or using matrices. Each method involves manipulating the equations to eliminate variables and find the values of the remaining variables.

4. Can you give an example of solving 3 simultaneous equations in 3D vector planes?

One example is the system of equations: x + y + z = 5, 2x - y + z = 3, and 3x + 2y - z = 1. This can be represented as the planes x + y + z = 5, 2x - y + z = 3, and 3x + 2y - z = 1 in 3D space. By manipulating the equations, the solution can be found as (x, y, z) = (1, 2, 2).

5. What are some real-world applications of 3 simultaneous equations and 3D vector planes?

Simultaneous equations and 3D vector planes have various real-world applications, such as in engineering, physics, and economics. For example, they can be used to model the motion of objects in space, analyze the forces acting on a structure, or determine the optimal production levels for a company.

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