16.1.1: Resources and Key Concepts

Prerequisite Topics

The following is a list of prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Linear Functions
  • Graphing Linear Functions: A direct prerequisite for solving systems of linear equations by graphing.
  • Linear Models: Required for working with applications of systems of linear equations.
  • Parallel and Perpendicular Lines: Understanding that parallel lines have the same slope is key to identifying inconsistent systems graphically.

Videos

• Systems of linear equations
  • Intermediate Algebra - An Introduction to Systems of Equations (Part A)
  • Intermediate Algebra - An Introduction to Systems of Equations (Part B)
  • Intermediate Algebra - An Introduction to Systems of Equations (Part C)
• Solving systems by substitution
  • Intermediate Algebra - Solving Small Systems of Linear Equations Algebraically (Part A)
• Solving systems by elimination
  • Intermediate Algebra - Solving Small Systems of Linear Equations Algebraically (Part B)

Definitions

• System of Equations: Two or more equations made up of two or more variables such that all equations in the system are considered simultaneously.
• System of Linear Equations (Linear System): A system consisting of two or more linear equations.
• Solution to a System: A combination of values for the variables that causes each equation in the system to become a true statement. For a 2x2 system, this is an ordered pair \((x,y)\).
• n-tuple: An ordered collection of \(n\) elements, used to represent solutions to systems with \(n\) variables.
• Consistent System: A system of equations that has one or more solutions.
• Inconsistent System: A system of equations that has no solutions. Graphically, this corresponds to parallel lines.
• Independent System: A consistent system with a finite number of solutions (for a 2x2 linear system, this means exactly one solution). Graphically, this corresponds to intersecting lines.
• Dependent System: A consistent system with an infinite number of solutions. Graphically, this corresponds to two identical lines (coincident lines).
• Substitution Method: A heuristic process for solving a system where one equation is solved for one variable, and that expression is then substituted into another equation in the system.
• Elimination Method: An algorithmic process for solving a system where two equations are added together to eliminate a variable. This often requires first multiplying one or both equations by a constant to ensure one variable has opposite coefficients.

Theorems

• Number of Solutions of Linear Systems: A consistent linear system has either exactly one solution or an infinite number of solutions. (This implies a linear system cannot have exactly two, three, etc., solutions).

Common Mistakes

• Solving for Only One Variable: After finding the value of the first variable (e.g., \(x\)), forgetting to substitute it back into an original equation to solve for the second variable.
• Errors in Substitution: When substituting an expression into another equation, making algebraic errors, especially with distribution if the expression is being multiplied by a coefficient.
• Errors in Elimination:
  • Incorrectly multiplying an entire equation by a constant (e.g., forgetting to multiply the constant term on the other side of the equals sign).
  • Making addition errors after multiplying, especially with negative numbers.
  • Multiplying by a constant that does not result in opposite coefficients for one of the variables.
• Assuming a Solution Exists: Concluding there is one solution when the system is actually inconsistent (parallel lines) or dependent (same line). Arriving at a false statement (e.g., \(9=13\)) indicates no solution. Arriving at a true identity (e.g., \(0=0\)) indicates infinite solutions.
• Incorrectly Stating the Solution for a Dependent System: For a dependent system with infinite solutions, simply writing "infinite solutions" is often insufficient. The solution should be expressed as an ordered pair in terms of one of the variables (e.g., \((x, -\frac{1}{3}x + \frac{2}{3})\)).
• Errors in Setting Up Systems from Word Problems: Misinterpreting the problem and creating incorrect linear equations to represent the given information.