Notes on Systems of Linear Equations

Overview of Systems of Linear Equations

• Systems of linear equations involve multiple equations with several variables considered simultaneously.

• The objective is to find numerical values for all variables that satisfy all equations simultaneously.

Types of Solutions

• There are three possible types of solutions for a system of linear equations:

  • Independent System: Exactly one solution exists; the equations represent two lines that intersect at a single point.

  • Dependent System: Infinitely many solutions exist; the equations represent the same line (coincident lines).

  • Inconsistent System: No solution exists; the equations represent parallel lines that never intersect.

Example of a System of Linear Equations

• Consider the following equations:

  1. $x + y = 15$

  2. $3x - y = 5$

• An example solution is the ordered pair $(4, 7)$:

  • Substitute into the first equation: $4 + 7 = 15$ (True)

  • Substitute into the second equation: $3(4) - 7 = 5$ (True)

Consistent vs. Inconsistent Systems

• Consistent Systems: Have at least one solution.

  • Independent: Exactly one solution (lines intersect).

  • Dependent: Infinitely many solutions (lines coincide).

• Inconsistent Systems: No solution exists (lines are parallel).

Verifying Solutions

To verify if an ordered pair is a solution, substitute the pair into each equation:

1. Substitute the ordered pair into both equations.

2. Check if both yield true statements.

For example, for the ordered pair $(5, 1)$ in the equations:

1. $x + 3y = 8$ Substitute: $5 + 3(1) = 8$ (True)

2. $2x - 9 = y$ Substitute: $2(5) - 9 = 1$ (True)

This confirms that $(5, 1)$ is indeed a solution to the system of equations.

Solving Systems by Graphing

• Graph both equations on a set of axes to visualize the system:

  • Techniques include using the y-intercept and slope to plot points or using intercepts.

  • Identify the type of system by examining the graphical representation:

  • If two lines intersect, it’s an independent system.

  • If lines are parallel, it’s inconsistent.

  • If lines overlap, it’s dependent.

Example of Graphing

• Solving the system:
  $2x + y = 8$
  $-x - y = 1$

• Solving yields:

  • Rearranging the first for y: $y = -2x + 8$

  • Rearranging the second for y: $y = -x - 1$

• After plotting, find the intersection point to identify the solution, e.g., at $(−3, -2)$ which confirms it is an independent system because the lines intersect at one point.

Practice and Review

• Use examples to practice identifying consistent vs. inconsistent systems.

• Graph various equations to visualize and determine types of solutions.

  • Examples include:

  1. $5 - 3y = -19$

  2. $4x + y = 11$

  3. $y = -3x + 6$

• Observe the intersection, coinciding, or divergent nature of lines for conclusions in solutions.