Systems of equations and inequalities Knowledge-Based Study

Study notes

1. What is a System of Equations?

   A system of linear equations consists of two or more linear equations sharing the same variables. A solution to a system is an ordered pair $(x, y)$ that satisfies all equations in the system simultaneously.

   Geometrically, each linear equation represents a straight line in the coordinate plane. The solution to the system corresponds to the intersection point of these lines.

   Classifying Linear Systems

   A system of two linear equations in two variables can have three possible types of solution sets:

   • Consistent and Independent: The lines intersect at exactly one point. The system has one unique solution $(x, y)$.
   • Inconsistent: The lines are parallel and never intersect because they share the same slope but have different $y$-intercepts. The system has no solution.
   • Consistent and Dependent: The equations represent the same line. The lines coincide completely. The system has infinitely many solutions.

   Solving Systems Algebraically

   While graphing provides visual intuition, algebraic methods yield exact coordinates without estimation.

   1. The Substitution Method

   The substitution method involves expressing one variable in terms of the other and substituting that expression into the second equation.

   • Step 1: Isolate one variable with a coefficient of 1 or $-1$ if possible.
   • Step 2: Substitute this expression into the other equation to obtain a single-variable equation.
   • Step 3: Solve for the variable, then back-substitute to find the remaining variable.

   For example, in the system:

   $y = 2x - 1$
   $3x + 2y = 12$

   Substitute $2x - 1$ for $y$ in the second equation: $3x + 2(2x - 1) = 12$, which simplifies to $7x - 2 = 12$, giving $x = 2$. Substituting back yields $y = 2(2) - 1 = 3$. The solution is $(2, 3)$.

   2. The Elimination (Addition) Method

   The elimination method aligns matching variable terms to eliminate one variable by adding or subtracting equations.

   • Step 1: Write both equations in standard form: $Ax + By = C$.
   • Step 2: Multiply one or both equations by non-zero constants so that the coefficients of one variable become opposites.
   • Step 3: Add the equations together to eliminate that variable, solve for the remaining variable, and substitute back to find the other coordinate.

   Graphing Linear Inequalities

   A linear inequality in two variables defines a half-plane bounded by a line. A system of linear inequalities consists of two or more inequalities whose solution is the intersection, or overlap, of their individual solution regions.

   Boundary Lines and Shading

   To determine the solution region of a system:

   • Boundary Lines: Graph the equation of each boundary line. Use a solid line for inequalities containing $\le$ or $\ge$, indicating the boundary is included. Use a dashed line for strict inequalities containing $<$ or $>$, indicating points on the line are not solutions.
   • Test Points: Choose a test point not on the boundary line, such as $(0, 0)$, to check which side of the boundary makes the inequality true. Shade the valid half-plane.
   • Feasible Region: The solution to the system is the region where all shaded areas overlap simultaneously. Points located in this intersection satisfy every inequality in the system.

   If there is no overlapping region between the shaded half-planes, the system has no solution.

   Interpreting Solution Regions and Boundary Intersections

   In many algebraic and real-world models, systems of equations and inequalities define boundaries and constraints.

   Corner Points (Vertices)

   The boundary intersections of a system of inequalities are called corner points or vertices. These vertices are found by solving the system of equations formed by the intersecting boundary lines.

   Recognizing Special Algebraic Outcomes

   When solving an algebraic system, the variables may eliminate completely:

   • A false numerical statement such as $0 = 7$ indicates an inconsistent system with no solution (parallel boundary lines).
   • A true identity such as $0 = 0$ indicates a dependent system with infinitely many solutions lying on the coincident line.

   Similarly, for inequalities, if parallel boundaries require shading in opposite, non-overlapping directions, the solution set is empty.