Systems of Equations and Matrices – Precalculus Study Guide

Ch. 9 – Systems and Matrices

Systems of Linear Equations

Systems of linear equations consist of two or more linear equations involving the same set of variables. The solution to a system is the set of values that satisfy all equations simultaneously.

• Single Equation: Represents a line on a graph. Every point on the line is a solution to the equation.

• System of Equations: Multiple equations, each representing a line. The solution is the point(s) where the lines intersect.

• Example: For the system $y = 2x + 3$ and $y = x + 4$, the intersection point is the solution to both equations.

Solving Systems of Linear Equations

There are several methods to solve systems of linear equations, including graphing, substitution, and elimination.

Substitution Method

• Definition: Solve one equation for one variable, then substitute into the other equation.

• Steps:

  1. Choose an equation and solve for $x$ or $y$.

  2. Substitute into the other equation.

  3. Solve for the remaining variable.

  4. Plug back to find the other variable.

  5. Check by substituting both values into both equations.

• Example: Solve $y = 7x - 14$ and $2x - y = 4$ by substitution.

Elimination Method

• Definition: Add or subtract equations to eliminate one variable.

• Steps:

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

  2. Multiply one or both equations so coefficients of one variable are equal (or opposites).

  3. Add or subtract equations to eliminate that variable.

  4. Solve for the remaining variable.

  5. Plug back to find the other variable.

  6. Check by substituting both values into both equations.

• Example: Solve $3x + 2y = 1$ and $-x + y = 3$ by elimination.

Multiplying Equations in Elimination Method

To eliminate a variable, you may need to multiply one or both equations by a factor so that the coefficients of the variable are equal or opposites.

If coefficients are...	Multiply...	Example
Equal, opposite signs	Nothing; just add	$7x + 12y = 17$ $-7x + 4y = 12$
Equal, same signs	Either eqn by $-1$	$7x + 12y = 17$ $7x + 4y = 12$
Factors of each other	Multiply one eqn	$4x + 2y = 7$ $2x + 5y = 4$
Anything else	Each eqn by a number	$4x + 2y = 7$ $3x + 5y = 4$

Classifying Systems of Linear Equations

Systems of linear equations can be classified based on the number of solutions:

Type	Description	Graph	Number of Solutions
Independent	Lines intersect at one point	Distinct intersection	1
Dependent	Lines are coincident (same line)	Overlap completely	Infinitely many
Inconsistent	Lines are parallel	No intersection	0

Introduction to Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to represent and solve systems of linear equations.

• Augmented Matrix: Represents a system of equations, including coefficients and constants.

• Example: The system $2x + (-3y) + z = -4$, $6x + 3y = 18$, $y - z = 8$ can be written as: $\begin{bmatrix} 2 & -3 & 1 & -4 \\ 6 & 3 & 0 & 18 \\ 0 & 1 & -1 & 8 \end{bmatrix}$

Matrix Row Operations

Row operations are used to solve systems of equations by transforming matrices.

Row Operation	System of Equations	Matrix	Notation
Swap two rows	Swap two equations	Swap two rows	$R_1 \leftrightarrow R_2$
Multiply row by a number	Multiply equation by a number	Multiply row by a number	$kR_1 \rightarrow R_1$
Add a multiple of one row to another	Add a multiple of one equation to another	Add a multiple of one row to another	$R_2 + kR_1 \rightarrow R_2$

Solving Systems Using Matrices (Row-Echelon Form)

To solve a system using matrices, use row operations to write the matrix in row-echelon form (upper triangular form).

• Row-Echelon Form: All zeros below the diagonal.

• Reduced Row-Echelon Form: All zeros below and above the diagonal, and leading coefficients are 1.

• Gaussian Elimination: The process of using row operations to solve systems.

• Gauss-Jordan Elimination: Extends Gaussian elimination to reduced row-echelon form.

Determinants & Cramer's Rule

Determinants are special numbers calculated from square matrices. Cramer's Rule uses determinants to solve systems of linear equations.

Determinant of a 2x2 Matrix

• Formula: For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $\det(A) = ad - bc$

• Example: $A = \begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix}$, $\det(A) = 3 \times 2 - 4 \times 5 = 6 - 20 = -14$

Determinant of a 3x3 Matrix

• Formula: For $A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$, $\det(A) = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)$

• Example: $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

Cramer's Rule

• Definition: A method for solving systems of linear equations using determinants.

• For 2 equations in 2 unknowns:

  • Given $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$

  • $x = \dfrac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}$

  • $y = \dfrac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}$

• For 3 equations in 3 unknowns:

  • Given $a_1x + b_1y + c_1z = d_1$, $a_2x + b_2y + c_2z = d_2$, $a_3x + b_3y + c_3z = d_3$

  • $x = \dfrac{\det(A_x)}{\det(A)}$, $y = \dfrac{\det(A_y)}{\det(A)}$, $z = \dfrac{\det(A_z)}{\det(A)}$

  • Where $A$ is the coefficient matrix, and $A_x$, $A_y$, $A_z$ are matrices with the respective column replaced by the constants.

Systems of Inequalities

Systems of inequalities involve finding the region of the graph that satisfies all inequalities in the system.

• Graphing Linear Inequalities:

  • Graph the boundary line (solid for $\leq$ or $\geq$, dashed for $<$ or $>$).

  • Shade the region that satisfies the inequality.

  • For systems, the solution is the overlapping shaded region.

• Graphing Nonlinear Inequalities:

  • Graph the boundary curve (e.g., parabola, circle).

  • Shade the region that satisfies the inequality.

• Example: Graph $y > 2x - 4$ and $x^2 + (y - 1)^2 \leq 9$.

Summary Table: Methods for Solving Systems

Method	When to Use	Steps
Graphing	Small systems, visual solutions	Graph each equation, find intersection
Substitution	Easy to isolate a variable	Solve for one variable, substitute
Elimination	Standard form, large coefficients	Multiply/add/subtract to eliminate variable
Matrices	Large systems, advanced methods	Write augmented matrix, use row operations
Cramer's Rule	Square systems, determinants nonzero	Calculate determinants, solve for variables

Additional info: These notes cover the essential concepts and methods for solving systems of equations and inequalities, as well as matrix operations and determinants, which are core topics in Precalculus Chapter 9.