Dependent System
Identify dependent systems of equations in Grade 10 algebra. Recognize when two equations represent the same line, producing infinitely many solutions, and write the solution set.
Key Concepts
Property A system with infinitely many solutions is a dependent system. The two lines coincide, meaning they are the same line. When you try to solve the system, you will get a true statement, such as $0 = 0$.
Solve $\begin{cases} 2x y = 3 \\ 6x + 3y = 9 \end{cases}$. Multiply the first equation by 3: $3(2x y) = 3(3)$ gives $6x 3y = 9$. Add the new equation to the second original equation: $(6x 3y) + ( 6x + 3y) = 9 + ( 9)$, which simplifies to $0 = 0$. Since $0=0$ is always true, the system has infinitely many solutions and is dependent.
Imagine you're solving a mystery with two clues. If you realize both clues are telling you the exact same thing, you don't have one answer—you have infinite possibilities that fit! That's a dependent system. The equations are just different ways of describing the same line, so every point on it is a solution.
Common Questions
What is a dependent system of equations?
A dependent system has infinitely many solutions because the two equations describe the same line. When solved algebraically, variables cancel and leave a true statement like 0 = 0.
How do you recognize a dependent system algebraically?
If one equation is a multiple of the other, the system is dependent. Trying to solve will yield a true statement (e.g., 0 = 0) with no specific solution — every point on the line is a solution.
How do you describe the solution set of a dependent system?
Write the solution as all points on the shared line: {(x, y) | y = 2x + 3} or parameterically as (t, 2t + 3) for all real t. There are infinitely many solutions.