Understanding Systems of Equations in Algebra

A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfy all equations simultaneously.

What is a System of Equations?

A system of equations consists of multiple equations that share variables.

The solution is the point(s) where all equations are true at the same time.

Types of Systems
  • Linear systems: All equations are linear.
  • Nonlinear systems: At least one equation is nonlinear.
Solving Linear Systems
  • Substitution: Solve one equation for a variable and substitute into the other.
  • Elimination: Add or subtract equations to eliminate a variable.
  • Graphical: Graph both equations and find the intersection point(s).
Number of Solutions
  • One solution: The lines intersect at a single point.
  • No solution: The lines are parallel.
  • Infinitely many solutions: The lines are coincident (the same line).
Practice Problems

  1. Solve: \( x + y = 5 \), \( x - y = 1 \)

  2. Solve: \( 2x + 3y = 7 \), \( x - y = 4 \)

  3. Graph the system: \( y = 2x + 1 \), \( y = -x + 4 \)

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