Question

Determine the values of r and s for which the given system of linear equations has (a) no solutions, (b) exactly one solution, and (c) infinitely many solutions?

Answer 1

To determine the values of r and s for a given system of linear equations, you need to analyze the number of solutions. If the lines are parallel, there is no solution. If the lines are overlapping, there are infinitely many solutions. If neither, the system has exactly one solution.

Step-by-step explanation:

In order to determine the values of r and s for which the given system of linear equations has:

(a) no solutions,

(b) exactly one solution,

(c) infinitely many solutions,

you need to solve the system of equations and analyze the resulting solution.

The detailed step-by-step explanation for each case can be found below:

Solution for (a): No solution

1. Write the given system of linear equations in the form of ax + by = c.
2. Compare the coefficients of x and y in the two equations to identify whether the lines are parallel or overlapping.
3. If the lines are parallel (a1/a2 = b1/b2 ≠ c1/c2), there is no solution.
4. If the lines are overlapping (a1/a2 = b1/b2 = c1/c2), there are infinitely many solutions.
5. If the lines are neither parallel nor overlapping, solve the system of equations to find the unique solution.

Solution for (b): Exactly one solution

1. Write the given system of linear equations in the form of ax + by = c.
2. Compare the coefficients of x and y in the two equations to identify whether the lines are parallel or overlapping.
3. If the lines are parallel, there is no solution.
4. If the lines are overlapping, there are infinitely many solutions.
5. If the lines are neither parallel nor overlapping (a1/a2 ≠ b1/b2), solve the system of equations to find the unique solution (x and y values).

Solution for (c): Infinitely many solutions

1. Write the given system of linear equations in the form of ax + by = c.
2. Compare the coefficients of x and y in the two equations to identify whether the lines are parallel or overlapping.
3. If the lines are parallel, there is no solution.
4. If the lines are overlapping (a1/a2 = b1/b2 = c1/c2), there are infinitely many solutions.
5. If the lines are neither parallel nor overlapping, solve the system of equations to find the unique solution.