The text from the image asks about systems of equations with no solution or infinitely many solutions and how the solution is determined by the constants in the equations. Here's the explanation:

When solving systems of linear equations, the relationships between the coefficients and constants determine whether the system has no solution, one unique solution, or infinitely many solutions:

1. No Solution (Inconsistent System):

   • This occurs when the equations represent parallel lines that never intersect. Mathematically, this happens if the ratios of the coefficients of the variables are equal, but the ratio of the constants is not: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
   • For example, $2x + y = 3$ and $4x + 2y = 6$ are parallel lines with no intersection.

2. Infinitely Many Solutions (Dependent System):

   • This occurs when the equations represent the same line. The coefficients and constants are proportional: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
   • For example, $x + y = 2$ and $2x + 2y = 4$ are essentially the same equation.

3. Unique Solution:

   • If neither of the above conditions is satisfied, the system has a unique solution (the lines intersect at one point).

Thus, the solution's nature depends on the ratios of coefficients and constants in the equations.

Let me know if you'd like more details or examples!

Here are 5 related questions to expand on this topic:

1. How do you determine if two equations are parallel or coincident geometrically?
2. Can a system of equations with three variables also have no solution or infinitely many solutions? Explain.
3. What happens when a system of equations is inconsistent in real-world applications?
4. How can substitution be used to identify an inconsistent system?
5. Are there any scenarios where substitution might not be the best method for solving a system of equations?

Tip: Always simplify equations to standard form (Ax + By = C) to make comparisons of coefficients and constants easier!