B5

Systems of Equations & Matrices

Introduction to Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The solution to a system is the set of variable values that satisfy all equations simultaneously. Systems of equations are fundamental in algebra and have many real-world applications, such as modeling cost, revenue, and constraints.

• Linear Equation: An equation of the form $ax + by = c$.

• System of Equations: A set of two or more equations with the same variables.

• Solution: The point(s) $(x, y)$ that satisfy all equations in the system.

Example: Solving for the intersection of two rental cost models.

Graphical Representation of Systems

Systems of equations can be represented graphically by plotting each equation as a line on the coordinate plane. The intersection point(s) of the lines represent the solution(s) to the system.

• One Solution: Lines intersect at a single point.

• No Solution: Lines are parallel and never intersect.

• Infinite Solutions: Lines are coincident (the same line).

Number of Solutions	Graphical Representation	Description
1 Solution	Intersecting Lines	Lines cross at one point
0 Solutions	Parallel Lines	Lines never meet
Infinite Solutions	Same Line	Lines overlap completely

Methods for Solving Systems of Linear Equations

There are several methods for solving systems of linear equations, each with its own advantages depending on the context.

Method 1: Substitution

• Solve one equation for one variable in terms of the other.

• Substitute this expression into the other equation.

• Solve for the remaining variable.

• Back-substitute to find the other variable.

Example: Given $y = 2x + 3$ and $x + y = 7$: Substitute $y$ into the second equation: $x + (2x + 3) = 7$ $3x + 3 = 7$ $3x = 4$ $x = \frac{4}{3}$ Then $y = 2(\frac{4}{3}) + 3 = \frac{8}{3} + 3 = \frac{17}{3}$

Method 2: Elimination (Addition/Subtraction)

• Write both equations in standard form $Ax + By = C$.

• Multiply one or both equations to align coefficients for one variable.

• Add or subtract equations to eliminate one variable.

• Solve for the remaining variable.

• Back-substitute to find the other variable.

Example: $2x + 3y = 8$ $4x - 3y = 2$ Add the equations: $(2x + 3y) + (4x - 3y) = 8 + 2$ $6x = 10$ $x = \frac{10}{6} = \frac{5}{3}$ Substitute $x$ into one equation to find $y$.

Method 3: Graphing

• Graph each equation on the same coordinate plane.

• Identify the intersection point(s).

• Check the coordinates of the intersection for the solution.

Note: Graphing is useful for visualizing solutions but may be less precise for non-integer intersections.

Piecewise Functions and Their Graphs

A piecewise function is defined by different expressions over different intervals of the domain. These are often used to model situations where a rule changes based on input value.

• Domain: The set of all input values for which the function is defined.

• Range: The set of all possible output values.

Example: $f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ 3x - 2 & \text{if } x \geq 2 \end{cases}$

To graph a piecewise function, plot each segment over its respective interval.

Application Problems Using Systems of Equations

Systems of equations are commonly used to solve real-world problems involving constraints and relationships between quantities.

• Cost Models: Comparing rental costs with different pricing structures.

• Test Scoring: Determining the number of questions of each type on an exam given total points and total questions.

• Geometry: Finding dimensions of a rectangle given perimeter and relationships between length and width.

Example: A test has 125 questions worth 1300 points. True/False questions are worth 8 points, Multiple Choice are worth 10 points. Let $T$ be the number of True/False questions and $M$ the number of Multiple Choice questions. $T + M = 125$ $8T + 10M = 1300$ Solve the system to find $T$ and $M$.

Practice Problems and Solution Types

Practice problems reinforce the concepts of solving systems by various methods and interpreting the number of solutions.

• Find the break-even point for cost and revenue functions.

• Graph systems and identify solution types (one, none, infinite).

• Set up and solve systems for geometric and application scenarios.

Summary Table: Solution Types for Systems of Two Linear Equations

Type	Graph	Algebraic Condition	Number of Solutions
Intersecting Lines	Lines cross	Different slopes	One
Parallel Lines	Lines never meet	Same slope, different intercepts	None
Coincident Lines	Lines overlap	Same slope and intercept	Infinite

Additional info: These notes include graphical and algebraic methods for solving systems, piecewise functions, and application problems, all central to College Algebra. Practice problems and worked examples reinforce the concepts and methods described.