Algebraic Expressions and Coordinate Geometry Study Guide

ALGEBRAIC EXPRESSIONS AND FACTORIZATION

Numerical expressions and variables are used to represent mathematical relationships. The following expressions illustrate basic algebraic combinations, powers, and distributive properties:

Use simplification and factoring for the expression: $18ab + 3a$ .
Identify terms in powers: $9m^4 - 9m^3$ .
Expanding and simplifying terms: $7ac^2 - 3b + 8b(3b - 2)$ . This involves applying the distributive property to the term $8b(3b - 2)$ .
Complex variable expression: $8ac^2(m - 1) + 12x(4n - m)$ .

LINEAR EQUATIONS IN VARIOUS FORMS

Linear equations define straight lines on a Cartesian plane and can be expressed in different formats. The slope-intercept form is defined as $y = mx + c$ , where $m$ represents the gradient and $c$ represents the y-intercept. Examples from the material include:

$y = -3x + 5$ : Here, the gradient ( $m$ ) is $-3$ and the y-intercept is $5$ .
$3x - 2y + 8 = 0$ : This is the standard form of a linear equation, which can be rearranged to slope-intercept form by solving for $y$ .
$y = -3x + 7$ : A line with a gradient of $-3$ and a y-intercept of $7$ .
$y = -3x$ : A line passing through the origin $(0, 0)$ with a gradient of $-3$ .

POINT VERIFICATION ON A LINEAR GRAPH

To determine if a specific coordinate point $(x, y)$ lies on a given line, the values of the coordinates must be substituted into the equation. For the equation $y = -\frac{2}{3}x + 3$ , evaluate the following points:

Point $(0, -3)$ : Substitute $x = 0$ and $y = -3$ into the equation. $y = -\frac{2}{3}(0) + 3 = 3$ . Since $3 \neq -3$ , the point $(0, -3)$ does not lie on the line.
Point $(3, 1)$ : Substitute $x = 3$ and $y = 1$ into the equation. $y = -\frac{2}{3}(3) + 3 = -2 + 3 = 1$ . Since $1 = 1$ , the point $(3, 1)$ lies on the line.

THE GRADIENT FORMULA AND SLOPE CALCULATION

The gradient ( $m$ ) of a line determines its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between two points $(x_1, y_1)$ and $(x_2, y_2)$ . The gradient formula is defined as:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Based on the points provided:

Point $A(-3, -3)$ , Point $B(5, -1)$ , and Point $C(10, 1)$ .
To calculate the gradient of the line passing through $A$ and $B$ : $m = \frac{-1 - (-3)}{5 - (-3)} = \frac{2}{8} = \frac{1}{4}$ .
To calculate the gradient of the line passing through $B$ and $C$ : $m = \frac{1 - (-1)}{10 - 5} = \frac{2}{5}$ .

DETERMINING EQUATIONS FROM GIVEN PARAMETERS

Equations can be constructed using specific properties such as the gradient and a point or given intercepts:

Gradient and Point: State the equation of a straight line with a gradient of $-5$ passing through the point $(0, 10)$ . Since the x-coordinate is $0$ , the y-coordinate $10$ is the y-intercept ( $c$ ). The resulting equation is $y = -5x + 10$ .
Two Points (Intercepts): A "drought line" passes through the points $(-4, 0)$ and $(0, 2)$ . First, calculate the gradient: $m = \frac{2 - 0}{0 - (-4)} = \frac{2}{4} = \frac{1}{2}$ . With a y-intercept ( $c$ ) of $2$ , the equation is $y = \frac{1}{2}x + 2$ .
Calculating Unknown Values: Given the equation $y = -10x - 1$ , find the value of $y$ when $x = 3$ . Calculation: $y = -10(3) - 1 = -30 - 1 = -31$ .

ANALYZING COORDINATE RELATIONSHIPS AND COLLINEARITY

Collinearity refers to a set of points that lie on the same straight line. This can be verified by checking if the gradients between consecutive points are equal. Given the points $A(-1, 7)$ , $B(0, 10)$ , and $C(1, 13)$ , perform the following calculations:

3.2.1 Gradient of AB: $m = \frac{10 - 7}{0 - (-1)} = \frac{3}{1} = 3$ .
3.2.2 Gradient of BC: $m = \frac{13 - 10}{1 - 0} = \frac{3}{1} = 3$ .
3.2.3 Analysis: What is noticed about points $A$ , $B$ , and $C$ is that they all lie on the same straight line because the gradients between the points are identical ( $m = 3$ ). This proves the points are collinear.

GRAPHICAL REPRESENTATION OF LINEAR FUNCTIONS

Graphing involves plotting lines based on their equations on a Cartesian plane:

$y = -3$ : This represents a horizontal line where all points have a constant y-value of $-3$ . It is parallel to the x-axis.
$x = 3$ : This represents a vertical line where all points have a constant x-value of $3$ . It is parallel to the y-axis.
$y = 2x - 3$ : This is a line with a positive gradient of $2$ and a y-intercept located at $(0, -3)$ .

PARALLEL LINES AND INTERCEPT ANALYSIS

Lines that are parallel to each other share the same gradient ( $m_1 = m_2$ ).

Parallel Equation Calculation: Find the equation of a line parallel to $y = 2x - 3$ that passes through the point $(1, 2)$ . Since the lines are parallel, the new line must have a gradient of $m = 2$ . Using the point-slope form: $y - 2 = 2(x - 1) \rightarrow y - 2 = 2x - 2 \rightarrow y = 2x$ .
Intercept Analysis: For a given straight line graph: - State the y-intercept: This is the point where the line crosses the y-axis (denoted as the point $(0, y)$ ). - State the x-intercept: This is the point where the line crosses the x-axis (denoted as the point $(x, 0)$ , which is found by setting $y = 0$ in the equation).