Precalculus Study Guide: Linear Equations, Matrices, and Systems of Equations

Linear Equations and Their Forms

Point-Slope and Slope-Intercept Forms

Linear equations can be expressed in several forms, each useful for different purposes. The point-slope form and slope-intercept form are commonly used to describe lines in the coordinate plane.

• Slope (m): The rate at which y changes with respect to x. For two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-Slope Form: Given a point $(x_1, y_1)$ and slope $m$: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: Expresses the line as: $y = mx + b$ where $b$ is the y-intercept.

• Example: For points $P = (2, -3)$ and $Q = (-6, 2)$: $m = \frac{2 - (-3)}{-6 - 2} = \frac{5}{-8} = -\frac{5}{8}$ Point-slope form: $y + 3 = -\frac{5}{8}(x - 2)$ Slope-intercept form: $y = -\frac{5}{8}x + b$ (solve for $b$ using a point)

Average Rate of Change

Definition and Calculation

The average rate of change of a function $f(x)$ between $x = a$ and $x = b$ measures how much $f(x)$ changes per unit change in $x$ over the interval.

• Formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

• Example: For $f(x) = x^2 - 2$ between $x = 2$ and $x = 5$: $f(2) = 2^2 - 2 = 2$ $f(5) = 5^2 - 2 = 23$ $\text{Average Rate of Change} = \frac{23 - 2}{5 - 2} = \frac{21}{3} = 7$

Matrices: Operations and Applications

Matrix Addition, Scalar Multiplication, and Subtraction

Matrices are rectangular arrays of numbers used to represent systems and perform operations. Common operations include addition, scalar multiplication, and subtraction.

• Addition/Subtraction: Matrices of the same size can be added or subtracted by combining corresponding elements.

• Scalar Multiplication: Multiply each entry of a matrix by a scalar.

• Example: Given $A = \begin{bmatrix} 9 & -3 \\ 4 & -4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}$, find $3A - 2B$: $3A = \begin{bmatrix} 27 & -9 \\ 12 & -12 \end{bmatrix}$ $2B = \begin{bmatrix} 2 & 8 \\ 0 & -2 \end{bmatrix}$ $3A - 2B = \begin{bmatrix} 25 & -17 \\ 12 & -10 \end{bmatrix}$

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second. The product is defined only when the number of columns in the first matrix equals the number of rows in the second.

• Example: For $A = \begin{bmatrix} 2 & -2 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -1 \\ 7 & 0 \end{bmatrix}$: $AB = \begin{bmatrix} (2 \times 2) + (-2 \times 7) & (2 \times -1) + (-2 \times 0) \\ (1 \times 2) + (2 \times 7) & (1 \times -1) + (2 \times 0) \end{bmatrix} = \begin{bmatrix} 4 - 14 & -2 + 0 \\ 2 + 14 & -1 + 0 \end{bmatrix} = \begin{bmatrix} -10 & -2 \\ 16 & -1 \end{bmatrix}$

Solving Systems of Equations

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It is applicable when the system has the same number of equations as unknowns and the determinant of the coefficient matrix is nonzero.

• General System: $\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$

• Determinants: $D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$ $D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}$ $D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}$

• Solutions: $x = \frac{D_x}{D}$ $y = \frac{D_y}{D}$

• Example: Solve $3x - 2y = 0$, $5x + 10y = 4$: $D = \begin{vmatrix} 3 & -2 \\ 5 & 10 \end{vmatrix} = 3 \times 10 - (-2) \times 5 = 30 + 10 = 40$ $D_x = \begin{vmatrix} 0 & -2 \\ 4 & 10 \end{vmatrix} = 0 \times 10 - (-2) \times 4 = 0 + 8 = 8$ $D_y = \begin{vmatrix} 3 & 0 \\ 5 & 4 \end{vmatrix} = 3 \times 4 - 0 \times 5 = 12 - 0 = 12$ $x = \frac{8}{40} = \frac{1}{5}$ $y = \frac{12}{40} = \frac{3}{10}$

Matrix Equations and Augmented Matrices

Systems of equations can be represented as matrix equations or augmented matrices, which are useful for systematic solution methods such as Gaussian elimination.

• Matrix Equation: $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable vector, and $B$ is the constant vector.

• Augmented Matrix: Combines the coefficient matrix and constant vector into one matrix for row operations. Example for system: $\begin{cases} x + 2y + 12z = 3 \\ y - 4x + 5z = -3 \\ -x - 2y + 7z = 1 \end{cases}$ Augmented matrix: $\begin{bmatrix} 1 & 2 & 12 & 3 \\ -4 & 1 & 5 & -3 \\ -1 & -2 & 7 & 1 \end{bmatrix}$

Summary Table: Matrix Operations

Operation	Description	Example
Addition	Add corresponding elements	$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$
Scalar Multiplication	Multiply each element by scalar	$2 \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$
Multiplication	Row by column multiplication	$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

Additional info:

• Some steps and explanations have been expanded for clarity and completeness.

• Matrix multiplication and Cramer's Rule are foundational for solving systems in Precalculus and Linear Algebra.