B5

Systems of Linear Equations

Introduction to Systems of 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 optimization problems.

• Linear Equation: An equation of the form $ax + by = c$, where $a$, $b$, and $c$ are constants.

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

Piecewise Defined Functions

A piecewise function is defined by different expressions depending on the input value. These functions are useful for modeling situations where a rule changes based on the domain.

• Example:

  • $f(x) = \begin{cases} 2x+1 & x < 2 \\ x^2-1 & x \geq 2 \end{cases}$

• Domain: All real numbers ($-\infty, \infty$)

• Range: $(-\infty, \infty)$

To graph a piecewise function, create a table of values for each piece and plot accordingly.

Writing Equations from Graphs

Given a graph, you can write the equation of a line by identifying the slope and y-intercept.

• Slope-intercept form: $y = mx + b$

• Example: A line passing through $(0,2)$ with slope $-2$ has equation $y = -2x + 2$.

Solving Systems of Linear Equations

Graphical Interpretation

Systems of two linear equations can have:

Number of Solutions	Graphical Representation
One Solution	Intersecting Lines
No Solution	Parallel Lines
Infinitely Many Solutions	Same Line (Coincident)

Methods for Solving Systems

There are several algebraic methods for solving systems of equations:

1. Substitution Method

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

• Substitute this expression into the other equation and solve for the remaining variable.

• Back-substitute to find the other variable.

• Example:

  • Given $\begin{cases} x + y = 4 \\ 2x - y = 1 \end{cases}$

  • Solve the first for $y$: $y = 4 - x$

  • Substitute into the second: $2x - (4 - x) = 1 \implies 3x = 5 \implies x = \frac{5}{3}$

  • Then $y = 4 - \frac{5}{3} = \frac{7}{3}$

2. Elimination (Addition) Method

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

• Multiply one or both equations by constants so that the coefficients of one variable are opposites.

• Add the equations to eliminate one variable, then solve for the other.

• Back-substitute to find the remaining variable.

• Example:

  • Given $\begin{cases} 3x + 2y = 12 \\ 2x - 2y = 4 \end{cases}$

  • Add: $(3x + 2y) + (2x - 2y) = 12 + 4 \implies 5x = 16 \implies x = \frac{16}{5}$

  • Substitute $x$ into one equation to find $y$.

3. Graphical Method

• Graph both equations on the same coordinate plane.

• The intersection point(s) represent the solution(s).

• If lines are parallel, there is no solution; if coincident, infinitely many solutions.

Classifying Solutions

• One Solution: Lines intersect at a single point.

• No Solution: Lines are parallel (same slope, different intercepts).

• Infinitely Many Solutions: Lines are coincident (same equation).

Applications of Systems of Equations

Word Problems and Modeling

Systems of equations are used to model and solve real-world problems, such as cost analysis, break-even points, and test scoring.

• Example 1: Car Rental Cost

  • Company A: $C_1(x) = 120 + 0.10x$

  • Company B: $C_2(x) = 150 + 0.08x$

  • To find when costs are equal, set $C_1(x) = C_2(x)$ and solve for $x$.

• Example 2: Test Scoring

  • Let $T$ = number of True/False questions, $M$ = number of Multiple Choice questions.

  • Given: $T + M = 125$ and $8T + 10M = 1300$

  • Solve the system to find $T$ and $M$.

Break-Even Analysis

• Break-even point: The value where cost equals revenue.

• Example: Given $C(x) = 0.85x + 350$, $R(x) = 1.1x$, set $C(x) = R(x)$ to find the break-even $x$.

Practice Problems

• Find the solution to a given system using substitution or elimination.

• Graph systems and interpret the number of solutions.

• Model real-world scenarios with systems of equations and solve for unknowns.

Summary Table: Types of Solutions for Systems of Two Linear Equations

Case	Graph	Number of Solutions
Intersecting Lines	Different slopes	One
Parallel Lines	Same slope, different intercepts	None
Coincident Lines	Same slope, same intercept	Infinitely many

Additional info: These notes also include step-by-step examples, graphical illustrations, and practice problems to reinforce understanding of solving systems of equations both algebraically and graphically.