Unit 3 Study Guide

Systems of Equations

Types of Solutions for Linear Systems

Systems of equations can be classified based on the number and type of solutions they possess. The graphical and algebraic approaches help identify these solutions.

• One Solution (Consistent and Independent): The lines intersect at a single point. The system is consistent and independent.

• No Solution (Inconsistent): The lines are parallel and never intersect. The system is inconsistent.

• Infinitely Many Solutions (Consistent and Dependent): The lines coincide (are the same line). The system is consistent and dependent.

Example: - $y = 2x + 1$ and $y = -x + 4$ intersect at $(1, 3)$ (one solution). - $y = 2x + 1$ and $y = 2x - 3$ are parallel (no solution). - $y = 2x + 1$ and $2y = 4x + 2$ are the same line (infinitely many solutions).

Solving Systems Graphically

Graph each equation and identify the intersection point(s) to find the solution(s).

• Plot both lines on the same coordinate plane.

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

Example: - $y = x + 2$ and $y = -x + 4$ intersect at $(1, 3)$.

Solving Systems Algebraically

There are two main algebraic methods: substitution and elimination.

• Substitution: Solve one equation for one variable and substitute into the other equation.

• Elimination: Add or subtract equations to eliminate one variable, then solve for the remaining variable.

Example (Substitution): $y = 2x + 1$ $x + y = 4$ Substitute $y$ into the second equation: $x + (2x + 1) = 4 \rightarrow 3x = 3 \rightarrow x = 1$, $y = 3$.

Example (Elimination): $2x + y = 7$ $x - y = 1$ Add equations: $(2x + y) + (x - y) = 7 + 1 \rightarrow 3x = 8 \rightarrow x = 8/3$, $y = 7 - 2x$.

Word Problems: Modeling with Systems

Set up equations based on the context and solve using algebraic methods.

• Define variables for unknowns.

• Write equations based on the problem statement.

• Solve the system using substitution or elimination.

Example: Let $x$ = number of $5 bills$, $y$ = number of $10 bills$. $x + y = 12$ $5x + 10y = 90$ Solve to find $x = 6$, $y = 6$.

Non-Linear Systems

Types of Solutions

Non-linear systems may involve quadratic equations and can have zero, one, or two solutions.

• No Solution: The curves do not intersect.

• One Solution: The curves touch at one point (tangent).

• Two Solutions: The curves intersect at two points.

Example: $y = x^2$ and $y = 2x + 3$ intersect at $(1, 5)$ and $(-3, -3)$.

Solving Non-Linear Systems

Use substitution or elimination to solve systems involving quadratic and linear equations.

• Substitute the expression for $y$ from the linear equation into the quadratic equation.

• Solve the resulting quadratic equation for $x$.

• Find corresponding $y$ values.

Example: $y = x^2$ $y = 2x + 3$ Set $x^2 = 2x + 3 \rightarrow x^2 - 2x - 3 = 0$ Factor: $(x - 3)(x + 1) = 0 \rightarrow x = 3, -1$ $y = 9, 1$

Matrices and Determinants

Definition of a Matrix

A matrix is a rectangular array of numbers arranged in rows and columns, enclosed by brackets. Each value is called an element.

• Matrix Dimension: Given by the number of rows and columns (e.g., $2 \times 3$ matrix).

• Element Notation: $a_{ij}$ refers to the element in row $i$, column $j$.

Determinants

The determinant is a scalar value that can be computed from a square matrix and is used in solving systems of equations.

• For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is:

$\det = ad - bc$

Example: $\begin{bmatrix} 7 & 2 \\ 3 & 1 \end{bmatrix}$ $\det = (7)(1) - (2)(3) = 7 - 6 = 1$

Cramer's Rule

Cramer's Rule provides a method for solving systems of linear equations using determinants.

• For the system $\begin{cases} ax + by = e \\ cx + dy = f \end{cases}$:

$x = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}$ $y = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}$

Example: $2x + 3y = 7$ $x - y = 1$ $\det = (2)(-1) - (3)(1) = -2 - 3 = -5$ $x = \frac{\begin{vmatrix} 7 & 3 \\ 1 & -1 \end{vmatrix}}{-5} = \frac{(7)(-1) - (3)(1)}{-5} = \frac{-7 - 3}{-5} = 2$ $y = \frac{\begin{vmatrix} 2 & 7 \\ 1 & 1 \end{vmatrix}}{-5} = \frac{(2)(1) - (7)(1)}{-5} = \frac{2 - 7}{-5} = 1$

3x3 Systems of Equations

Types of Solutions

Systems with three variables can have one solution, no solution, or infinitely many solutions, represented by the intersection of three planes.

• One Solution: All three planes intersect at a single point.

• No Solution: The planes do not all intersect at a single point (e.g., parallel planes).

• Infinitely Many Solutions: The planes intersect along a line or coincide.

Example: $x + y + z = 13$ $2x - y + 3z = 19$ $x + 2y - z = 7$ Solution: $(4, 5, 4)$

Solving 3x3 Systems Algebraically

Use elimination or substitution to reduce the system to two equations in two variables, then solve.

• Eliminate one variable by combining two equations.

• Solve the resulting $2 \times 2$ system.

• Back-substitute to find the third variable.

Example: $x + y + z = 6$ $2x - y + 3z = 14$ $x + 2y - z = 4$ Eliminate $z$ and solve for $x$ and $y$, then find $z$.

Word Problems with 3x3 Systems

Set up three equations based on the context and solve using algebraic methods.

• Define three variables for unknowns.

• Write three equations based on the problem statement.

• Solve using elimination or substitution.

Example: Let $x$, $y$, $z$ be the number of adults, teens, and children at an event. $x + y + z = 200$ $5x + 3y + 2z = 370$ $x = 2z$ Solve for $x$, $y$, $z$.

Tables

Matrix Dimensions Table

Rows	Columns	Matrix Name
2	3	2 x 3 Matrix
3	2	3 x 2 Matrix
2	2	2 x 2 Matrix

Types of Solutions Table

Type	Graphical Representation	Number of Solutions
Consistent & Independent	Intersecting lines	1
Inconsistent	Parallel lines	0
Consistent & Dependent	Coinciding lines	Infinitely many