Final answer:
The value of k for which the system of equations has exactly one solution is any value except -44.06. This ensures the slopes of the lines are different and therefore not parallel.
Step-by-step explanation:
The student asks for the values of k for which the system of linear equations 13x − 17y = 10 and kx + 19y = −29 has exactly one solution. For a system of linear equations to have exactly one solution, the equations must be independent and not multiples of each other, meaning the lines they represent are not parallel. This can be ensured when the coefficients of x and y in the two equations have a different ratio, i.e., the slopes of the two lines are different.
A linear equation has the form y = a + bx, where a is the y-intercept and b is the slope. In the first equation, the coefficient of x is 13 and that of y is −17, giving it a slope of −13/17. For the second equation to have a different slope, and thus for the system to have exactly one solution, the ratio of k to 19 must not equal the ratio of 13 to −17. Therefore, k must not equal −13 * 19/17 or −44.06. Any other value for k would result in the system having exactly one solution. In summary, k ≠ −44.06 leads to a unique solution.