In algebra, determine the value of r for which the linear equation 13x + 5 = r x + 18 has no solution (the lines are parallel and distinct), and then choose the correct value of r from the options.

Introduction / Context:
This question focuses on when a linear equation in one variable has no solution. When we compare two linear expressions in x, there will be no solution if the coefficients of x are equal but the constant terms are different. In such a case the equation becomes a contradiction like 5 = 18. We must choose the value of r that produces this situation.

Given Data / Assumptions:

• The equation is 13x + 5 = r x + 18.
• x is a real variable.
• We seek a value of r such that the equation has no solution.
• All algebraic operations are valid over real numbers.

Concept / Approach:
Rearrange the equation into standard form by bringing all x terms to one side and constants to the other. If the coefficient of x becomes zero while the constant term remains non zero, the equation has no solution. That happens precisely when the coefficients of x on both sides are equal but the constant terms differ.

Step-by-Step Solution:
Start with 13x + 5 = r x + 18. Subtract r x from both sides: 13x - r x + 5 = 18. Combine like terms: (13 - r)x + 5 = 18. Subtract 5 from both sides: (13 - r)x = 13. For the equation to have no solution, the coefficient of x must be zero while the right hand side is non zero. Thus set 13 - r = 0. Solve 13 - r = 0 to get r = 13. Then the equation becomes 0·x = 13, which is impossible for any real x.
Verification / Alternative check:
Substitute r = 13 directly into the original equation: 13x + 5 = 13x + 18. Subtract 13x from both sides to obtain 5 = 18, which is a contradiction, proving there is no solution. Any other value of r would leave a non zero coefficient of x and produce a unique solution.

Why Other Options Are Wrong:
Options a, c, d, and e produce coefficients 13 - r that are not zero, so the resulting equation has exactly one solution. For example, if r = 5, we obtain 8x = 13, which has the solution x = 13/8. Therefore these values of r do not create a contradiction.

Common Pitfalls:
A common error is to think that equal constants guarantee no solution, while in fact it is equal coefficients with different constants that cause inconsistency. Another pitfall is forgetting to collect like terms correctly and miscomputing 13 - r, leading to incorrect conclusions about when the coefficient of x vanishes.

Final Answer:
The equation has no solution when r = 13.