The image you uploaded contains a true/false problem with the following three statements:

1. Statement 1: If the number of rows of an augmented matrix in reduced row-echelon form is greater than the number of columns (to the left of the vertical bar), then the system has infinitely many solutions.

   • Explanation: This statement is False. For a system of linear equations, if the number of rows (equations) is greater than the number of columns (variables), it does not necessarily imply that the system has infinitely many solutions. Instead, this suggests that the system might be inconsistent (having no solution) if the matrix leads to contradictions. Infinite solutions arise typically when there are more variables than equations and no contradictions occur.

2. Statement 2: If the solution to a system of linear equations is given by (4 - 2z, -3 + z, z), then (4, -3, 0) is a solution to the system.

   • Explanation: This statement is True. Substituting $z = 0$ into the parametric solution $(4 - 2z, -3 + z, z)$ yields $(4, -3, 0)$, confirming that this is a solution when $z = 0$.

3. Statement 3: If a matrix is in reduced-row echelon form, then the first nonzero entry in each row is a 1, and has 0's below it.

   • Explanation: This statement is False. While it is true that the first nonzero entry in each row (the pivot) is 1, reduced row-echelon form (RREF) requires that the pivot must have 0's both above and below it. The statement only mentions 0's below it, which is a condition for row-echelon form but not reduced row-echelon form.

Summary of Answers:

1. False
2. True
3. False

Would you like more details or further clarifications?

Here are 5 related questions to expand your knowledge:

1. What conditions guarantee that a system of linear equations has no solutions?
2. What is the difference between row-echelon form and reduced row-echelon form?
3. How do you solve a system of equations using augmented matrices and Gaussian elimination?
4. What are the necessary conditions for a system of equations to have infinitely many solutions?
5. How do parametric solutions for systems of linear equations work?

Tip: In reduced row-echelon form, the matrix is fully simplified, meaning that each leading 1 must also have zeros above it (in addition to zeros below).