Consider a system 
A
x
=
b
Ax=b of 
4
4 linear equations in 
5
5 variables. Which of the following statements is true?

The system 
A
x
=
b
Ax=b is consistent if and only if 
b
b is a linear combination of the 
5
5 columns of 
A
A.

The system 
A
x
=
b
Ax=b is consistent if and only if 
A
A is a linear combination of the column of 
b
b.

The system 
A
x
=
b
Ax=b is consistent if and only if 
A
A is a linear combination of the 
5
5 columns of 
b
b.

The system 
A
x
=
b
Ax=b is consistent if and only if 
b
b is a linear combination of the 
4
4 columns of 
A
A.

The correct statement is:

**The system \( A x = b \) is consistent if and only if \( b \) is a linear combination of the 4 columns of \( A \).**

Here’s why:

In a system of linear equations \( A x = b \), the matrix \( A \) has dimensions \( 4 \times 5 \) (4 rows, 5 columns), meaning it represents 4 equations in 5 variables. For the system to be consistent, the vector \( b \) (the right-hand side) must lie in the column space of the matrix \( A \). The column space of \( A \) is spanned by its 4 columns, meaning that \( b \) must be expressible as a linear combination of those 4 columns for the system to have at least one solution.

Thus, the last option is correct:  
**"The system \( A x = b \) is consistent if and only if \( b \) is a linear combination of the 4 columns of \( A \)."**

Would you like a further explanation on the consistency of linear systems?

Here are 5 related questions to expand your understanding:
1. What is the column space of a matrix?
2. How can we determine if a system of linear equations is consistent?
3. What does it mean for a vector to be a linear combination of other vectors?
4. How does the number of variables affect the number of solutions in a system of linear equations?
5. What is the rank of a matrix, and how does it relate to the consistency of a system?

**Tip:** When dealing with a system of linear equations, always check if the right-hand side vector lies in the column space of the coefficient matrix to determine consistency.

Consider a system Ax = b of 4 linear equations in 5 variables. Which of the following statements is true?

1. The system Ax = b is consistent if and only if b is a linear combination of the 5 columns of A.

2. The system Ax = b is consistent if and only if A is a linear combination of the column of b.

3. The system Ax = b is consistent if and only if A is a linear combination of the 5 columns of b.

4. The system Ax = b is consistent if and only if b is a linear combination of the 4 columns of A.

Understand how to determine the consistency of the system Ax = b with 4 linear equations and 5 variables. Learn about the importance of linear combinations of vectors, column space, and the conditions for a consistent system in linear algebra.

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