Defining a System of Linear Equations
A system of linear equations is a set of two or more linear equations sharing the same variables, where a solution is an ordered pair (x, y) satisfying every equation simultaneously. Grade 11 students in enVision Algebra 1 (Chapter 4: Systems of Linear Equations and Inequalities) learn to write systems using brace notation grouping the equations together. Graphically, the solution is the point where the lines intersect. Systems can have one solution, no solution (parallel lines), or infinitely many solutions (same line).
Key Concepts
Property A system of linear equations is a set of two or more linear equations that share the same variables. A solution to the system is an ordered pair $(x, y)$ that satisfies every equation in the system. The general form for a system of two linear equations is: $$ \begin{cases} A 1x + B 1y = C 1 \\ A 2x + B 2y = C 2 \end{cases} $$.
Examples A system with variables $x$ and $y$: $$ \begin{cases} y = 2x + 1 \\ y = x + 4 \end{cases} $$ A system in standard form: $$ \begin{cases} 3x 2y = 8 \\ x + 5y = 3 \end{cases} $$.
Explanation A system of linear equations consists of multiple linear equations considered together. The goal is to find a single point, or a set of points, that makes all equations in the system true simultaneously. Graphically, this corresponds to finding the point(s) where the lines representing each equation intersect. The brace symbol is used to group the equations together, indicating they form a system.
Common Questions
What is a system of linear equations?
A system of linear equations is two or more linear equations with the same variables, considered together. The goal is to find ordered pairs that satisfy all equations simultaneously.
What does a solution to a system look like?
A solution is an ordered pair (x, y) that makes every equation in the system true when substituted in.
How are systems of equations represented graphically?
Each equation graphs as a line. The solution is the point (or points) where all lines intersect.
How many solutions can a system of two linear equations have?
Exactly one solution (lines intersect at one point), no solution (parallel lines), or infinitely many solutions (the equations represent the same line).
What notation is used to write a system of linear equations?
A brace symbol groups the equations together, indicating they are considered simultaneously, for example: {A₁x + B₁y = C₁, A₂x + B₂y = C₂}.
What are the three methods for solving a system of linear equations?
Graphing (find intersection), substitution (solve one equation for a variable and substitute), and elimination (add/subtract equations to cancel one variable).