Systems of Equations

Multiply equation (2) by −2:

Add equation (1) and (2'):

Substitute into equation (1):

Solution:

Geometrically, these two lines intersect at the point .

From equation (1), .
Substitute this into equation (2):

This is a contradiction. Therefore, the system has no solution.

Geometrically, these are parallel and distinct lines.

Multiply equation (1) by −3:

Add equation (1') and (2):

This is always true. The system has infinitely many solutions.
To express the solution, let , where is any real number (a parameter).
From equation (1), .
Solution:

Geometrically, these are coincident lines.

Step 1: Eliminate using (1) and (2).
Add (1) and (2):

Step 2: Eliminate using (1) and (3).
Multiply (1) by −2:

Add (1') and (3):

Step 3: Solve the 2x2 system (4) and (5).

From (5), . Substitute into (4):

Substitute into :

Step 4: Substitute into an original equation (e.g., (1)) to find .

Solution:

Geometrically, these three planes intersect at the point .

The equation is a contradiction. This means there is no set of values that can satisfy all three original equations simultaneously. Therefore, the system has no solution.

Geometrically, this means the three planes do not have any common point of intersection. Possible configurations include:

1. All three planes are parallel and distinct.
2. Two planes are parallel and distinct, and the third plane intersects them.

+. The planes intersect in pairs forming parallel lines (a "triangular prism" shape with no common interior).

1. Two planes are coincident, and the third plane is parallel and distinct to them.

We can use elimination or substitution. Let's use elimination.
Multiply equation (1) by 3 to make the y-coefficients opposites:

Add equation (1') and equation (2):

Substitute into equation (1):

Check in equation (2): . This is correct.

Answer: The solution is .

Use elimination.
Step 1: Eliminate using equations (1) and (2).
Add (1) and (2):

Step 2: Eliminate using equations (1) and (3).
Multiply equation (1) by 3:

Add (1') and (3):

Step 3: Solve the system of equations (4) and (5).

Notice that equation (5) is equation (4): .
Since they are multiples, they represent the same line in the -plane (if we were thinking in 2D). This means there are infinitely many solutions for and that satisfy both.

To express the solution, let , where is a parameter.
Substitute into equation (4):

Now, substitute and back into one of the original equations (e.g., (1)) to find .

The solution is:

This can be written in vector form as .

Geometric Interpretation: The three planes intersect in a line. The equation found is the parametric equation of this line of intersection.

Answer: Infinitely many solutions, representing a line: .

Step 1: Eliminate using (1) and (3).
Add (1) and (3):

Step 2: Substitute into equations (1) and (2).

• Into (1): (4)
• Into (2): (5)
• Into (3): (6)

Notice that equation (4) is , and equation (6) is . These are equivalent: multiply (6) by −1 to get . So equations (4) and (6) are dependent. We essentially have the system:

• (from (1) and (3) combined effectively)
  
• (from (2) after subbing )

We need to solve this system for and : .
Substitute this into :

Case 1: , i.e., .
Then .
If , then .
And we already found .
So, if , the system has a unique solution: .

Case 2: , i.e., .
Substitute into :

This equation is , which is true for any value of . This indicates infinitely many solutions when .
If , we have . The equations for and become dependent:

• (from (4) or (6))

• (from (5) with ).

  • This is , so it's the same as .

Let . Then . So, if , the solutions are . This is a line.

Summary of findings:

• No solution?
  From :
  

  • If (i.e. ), we get , which is infinitely many solutions.
  • If , we get , a unique solution.

  It seems there is no case for "no solution" based on this reduction. Let's double check.
  The initial reduction came from (1) and (3). This part is always true.
  The system then reduced to:

  If these two lines (in variables ) are parallel and distinct, we'd get no solution.
  Their "slopes" (if is like function of ):
  From (slope 1)
  From .
  

  • If , . (slope )
    For them to be parallel, .
    If , then . Both equations become .
    They are coincident, leading to infinite solutions.
  • What if ?

  The system is:

  Substitute into

  And .
  So if (which is ), we get unique solution .
  This fits case.