System of equations
Understand systems of equations in Grade 10 algebra. Define systems of two or more equations with shared variables, identify solution types, and set up linear systems from real-world problems.
Key Concepts
A system of equations is a collection of two or more equations containing two or more of the same variables. A linear system contains two linear equations in two like unknowns. Solutions of systems are ordered pairs, representing the point of intersection.
The system $y = 3x 2$ and $y = x + 6$ has two equations with variables $x$ and $y$. The solution is where they meet. For variables $a$ and $b$, a system could be $2a + b = 7$ and $a b = 2$. A cost system might be $C = 25 + 5d$ and $C = 10d$, where $d$ is days and $C$ is cost.
Think of it like a treasure hunt with two different maps! Each equation is a map drawing a line. The treasure, or the solution, is the one spot where the lines cross. You're looking for the single coordinate pair $(x, y)$ that makes both equations true. It's the ultimate point of agreement between two mathematical stories.
Common Questions
What is a system of equations?
A system of equations is a collection of two or more equations with the same variables. A solution satisfies all equations simultaneously. Linear systems involve linear equations in two or more variables.
What are the three possible solution types for a system of two linear equations?
One solution (lines intersect at one point), no solution (parallel lines), or infinitely many solutions (same line). These correspond to consistent independent, inconsistent, and consistent dependent systems.
How do you set up a system of equations from a word problem?
Assign variables for each unknown, write one equation per condition given in the problem, then solve. Two unknowns require two equations; three unknowns require three equations.