Dependent System
A dependent system of linear equations consists of two equations that represent the exact same straight line. Because the lines coincide completely, the system has infinitely many solutions. It is a specific type of consistent system.
Key Concepts
Graphically, the two linear equations plot as coincident lines, meaning one line lies exactly on top of the other.
Algebraically, the condition for a dependent system is that the ratios of their corresponding coefficients and constant terms are strictly equal.
Any coordinate pair (x, y) that satisfies the first equation will automatically satisfy the second equation.
Formula & Equation
a1/a2 = b1/b2 = c1/c2
a1, b1 and a2, b2 are the coefficients of the variables x and y respectively, while c1 and c2 are the constant terms of the two linear equations.
Common Misconceptions
Myth: Dependent systems and inconsistent systems are the same thing because neither has a single unique solution.
Fact: They are fundamentally different. A dependent system has infinitely many solutions because the lines overlap, making it a consistent system. An inconsistent system has zero solutions because the lines are parallel.
Real World Applications
Purchasing 2 notebooks and 3 pens for 50 rupees, and later purchasing 4 notebooks and 6 pens for 100 rupees. Both statements represent the exact same cost relationship.
Mixing ingredients where a recipe calls for 1 cup of sugar to 2 cups of flour, and another calls for 2 cups of sugar to 4 cups of flour. The ratio remains identical.
Frequently Asked Questions
What is the difference between an independent and a dependent system?
An independent system has exactly one unique solution where two lines intersect at a single point, whereas a dependent system has infinitely many solutions because the lines coincide.
How do you solve a dependent system of linear equations?
When you try to solve it using substitution or elimination, both variables will cancel out, leaving a true statement like 0 = 0, which indicates there are infinitely many solutions.