Question

y = 3x + 5 y = ax + b What values for a and b make the system inconsistent? What values for a and b make the system consistent and dependent? Explain.

Answer 1

• inconsistent: a=3, b≠5
• dependent: a=3, b=5

Explanation:

Given the following system of equations, you want to know values of 'a' and 'b' that (i) make the system inconsistent, and (ii) make the system consistent and dependent.

• y = 3x +5
• y = ax +b

(i) Inconsistent

The system is inconsistent when it describes lines that are parallel and have no point of intersection. A solution to one of the equations cannot be a solution to the other.

Parallel lines have the same slope, but different y-intercepts. The system will be inconsistent when a=3 and b≠5.

(ii) Consistent, dependent

The system is consistent when a solution to one equation can be found that is also a solution to the other equation. The system is dependent if the two equations describe the same line (there are infinitely many solutions).

Here, the y-coefficients are the same in both equations, so the system will be dependent only if the values of 'a' and 'b' match the corresponding terms in the first equation:

The system is dependent when a=3, b=5.

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Additional comment

Dependent systems are always consistent.

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