9.1 Systems of Linear Equations: Two Variables

Linear equations are the building blocks of algebra, and solving systems of them is a crucial skill. This topic dives into various methods for tackling these systems, from graphing to substitution and elimination.

Understanding how to solve linear systems opens doors to modeling real-world problems. We'll explore how to determine if systems have one, no, or infinite solutions, and how to express those solutions mathematically.

Solving Systems of Linear Equations with Two Variables

Graphing solutions of linear systems

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• A system of linear equations (also known as simultaneous equations) contains two or more linear equations with the same variables
  • Each equation represents a line on a coordinate plane (e.g., $2x + 3y = 6$ and $x - y = 1$)
  • The solution to the system is the point(s) where the lines intersect (e.g., $(1, 2)$)
• Graphing both equations on the same coordinate plane helps visually determine the solution
  • Lines intersecting at a single point indicate one unique solution (consistent and independent system)
  • Parallel lines that do not intersect indicate no solution (inconsistent system)
  • Coincident lines (same line) indicate infinitely many solutions (consistent and dependent system)
• The coordinates of the point of intersection satisfy both equations simultaneously (e.g., $x = 1$ and $y = 2$)

Substitution and elimination methods

• Substitution method involves solving one equation for a variable and substituting it into the other equation
  1. Solve one equation for one variable in terms of the other (e.g., $y = 2x - 1$)
  2. Substitute the expression into the other equation (e.g., $2x + 3(2x - 1) = 6$)
  3. Solve the resulting equation for the remaining variable (e.g., $x = 1$)
  4. Substitute the solved variable's value back into the expression from step 1 to find the other variable's value (e.g., $y = 2(1) - 1 = 1$)
• Elimination method involves manipulating the equations to eliminate one variable
  1. Multiply equations by constants to make one variable's coefficients equal in magnitude but opposite in sign (e.g., multiply $x - y = 1$ by 2 to get $2x - 2y = 2$)
  2. Add the equations together to eliminate one variable (e.g., $(2x + 3y = 6) + (2x - 2y = 2)$ becomes $4x + y = 8$)
  3. Solve the resulting equation for the remaining variable (e.g., $y = 2$)
  4. Substitute the solved variable's value into one of the original equations to find the other variable's value (e.g., $x - 2 = 1$, so $x = 3$)

Consistency of linear systems

• A system of linear equations can be classified as:
  • Consistent and independent: One unique solution (intersecting lines)
  • Inconsistent: No solution (parallel lines)
  • Consistent and dependent: Infinitely many solutions (coincident lines)
• Determine the type of system by:
  • Graphing the equations and observing the lines' relationship
  • Using substitution or elimination to solve the system and interpreting the results
    • A unique solution indicates a consistent and independent system
    • A contradiction (e.g., $0 = 1$) indicates an inconsistent system
    • An identity (e.g., $0 = 0$) indicates a consistent and dependent system

Parametric form for dependent systems

• Consistent and dependent systems have infinitely many solutions
• Parametric form expresses the solution set using a parameter, typically $t$
• To express the solution in parametric form:
  1. Solve one equation for one variable in terms of the other (e.g., $y = 2x$)
  2. Replace the other variable with the parameter $t$ (e.g., $x = t$, so $y = 2t$)
  3. The solution set is represented as $(x, y) = (t, 2t)$, where $t$ is any real number

Real-world applications of linear systems

• Real-world problems can be modeled using systems of linear equations with two variables (e.g., supply and demand, cost analysis, mixture problems)
• To solve real-world problems:
  1. Identify unknown quantities and assign variables (e.g., let $x$ be the number of adults and $y$ be the number of children)
  2. Write a system of linear equations representing the relationships between variables based on given information (e.g., total number of people: $x + y = 100$, total ticket cost: $10x + 5y = 750$)
  3. Solve the system using substitution, elimination, or graphing (e.g., $x = 50$ and $y = 50$)
  4. Interpret the solution in the context of the original problem, considering constraints or limitations (e.g., there were 50 adults and 50 children)

Advanced solution methods

• Augmented matrix: A method of representing a linear system in matrix form, combining the coefficient matrix with the constant terms
• Gaussian elimination: A systematic approach to solving linear systems by transforming the augmented matrix into row echelon form
• Cramer's rule: A method for solving linear systems using determinants, particularly useful for systems with a unique solution