Let's analyze and solve the given questions step by step:

Part A:

The given equation is:

$4x + 3 = 4(1 + x) + 1$

Sherron receives an error message when she inputs this equation. Let’s analyze why:

1. Expand and simplify the equation:

Expand the right-hand side: $4x + 3 = 4(1) + 4(x) + 1$ $4x + 3 = 4 + 4x + 1$

Simplify the constants: $4x + 3 = 4x + 5$

2. Isolate terms:

Subtract $4x$ from both sides: $3 = 5$

This is a contradiction because 3 is not equal to 5. The equation has no solution. This is called an inconsistent equation, and a properly programmed solver would return an error message indicating that there is no solution.

Explanation: Sherron gets an error message because the equation is inconsistent—it has no solution. This occurs when the two sides of the equation contradict each other after simplification.

Part B:

The given equation is:

$5 + 6y = 9y + 2 + 3(1 - y)$

Let’s solve it step by step:

1. Expand and simplify:

Expand $3(1 - y)$ on the right-hand side: $5 + 6y = 9y + 2 + 3 - 3y$

Simplify the constants on the right-hand side: $5 + 6y = 9y + 5 - 3y$

Simplify further: $5 + 6y = 6y + 5$

2. Isolate terms:

Subtract $6y$ from both sides: $5 = 5$

This is always true, regardless of the value of $y$. This means the equation is an identity and is true for all values of $y$.

Result and Explanation: The result is that the equation is an identity—it is true for all values of $y$. A properly programmed solver should indicate that there are infinitely many solutions.

Summary of Results:

• Part A: Sherron’s program shows an error because the equation has no solution (it is inconsistent).
• Part B: The equation has infinitely many solutions (it is an identity).

Would you like further clarification or details on these solutions?

Related Questions:

1. What are the steps to determine if a linear equation has no solution?
2. How do you identify when an equation has infinitely many solutions?
3. Can an equation have exactly one solution, no solution, or infinitely many solutions? How do we decide which case applies?
4. Why do contradictions like $3 = 5$ occur in inconsistent equations?
5. How can a program be designed to handle equations with no solution or infinitely many solutions?

Tip: Always simplify both sides of the equation fully before deciding whether it is consistent, inconsistent, or an identity.