Unit 3 Study Guide: Pre-Linear Algebra, Equations, and Inequalities

Unit 3: Pre-Linear Algebra – Competences and Outline

• Key Competence: By the conclusion of this unit, students should possess the ability to solve mathematical problems involving linear equations and inequalities, including the ability to represent these solutions graphically.
• Core Unit Outline:
  • Linear functions
  • Equations
  • Inequalities

3.1 Linear Functions: Definitions and Forms

• Definition of a Linear Function: A linear function describes a specific relationship between two variables (conventionally denoted as $x$ and $y$) such that their graphical representation is a straight line.
• Standard Equation Form: A linear function is expressed in the form $y = mx + c$.
  • $m$: Represents the slope or gradient of the line.
  • $c$: Represents the $y$-intercept.
  • Both $m$ and $c$ function as constants.
• General Examples:
  • $y = x + 1$
  • $y = 2x - 3$
  • $y = -3x + 4$
• Special Cases:
  • Horizontal Lines: Occur when the value of $y$ is constant, expressed as $y = c$. Examples provided include $y = 3$ and $y = -\frac{3}{2}$.
  • Vertical Lines: Graphical representations where the $x$ value remains constant, such as $x = \frac{3}{2}$ and $x = -2$.

3.1.1 - 3.1.2 The Foundation: The Number Line

• Role of the Number Line: It serves as a tool for locating and comparing rational numbers relative to a central point, zero ($0$).
• Structure and Directions:
  • Origin: The point represented by the number $0$.
  • Positive Numbers: Situated to the right of zero.
  • Negative Numbers: Situated to the left of zero.
• Coordinates on a Line: Every point on a number line corresponds to a single number representing its distance and direction from the origin. This value is called the coordinate.
  • Verbatim Examples from Figure 3:
    • Point $A$ is $4$ units to the right; represented as $A(4)$.
    • Point $B$ is $3$ units to the left; represented as $B(-3)$.
    • Point $C$ is at $\frac{1}{2}$; represented as $C(\frac{1}{2})$.
    • Point $D$ is at $-5$; represented as $D(-5)$.

3.1.3 Spatial Positioning: Points on a Plane Surface

• Requirement for Localization: Placing a point on a flat plane requires two numbers (an ordered pair) rather than one, indicating position relative to a reference system.
• The Ordered Pair Concept:
  • Expressed as $(x, y)$.
  • The first number denotes the horizontal position (e.g., a column).
  • The second number denotes the vertical position (e.g., a row).
• Importance of Sequence: The order is critical because $(3, 5) \neq (5, 3)$.
• Desk Arrangement Metaphor: In a classroom grid, a shaded desk in the third column and fifth row is identified by the ordered pair $(3, 5)$.

3.1.4 - 3.1.5 Drawing and Labeling Axes

• Reference System: To locate points accurately, two perpendicular lines are used: one horizontal and one vertical. They meet at a point called the Origin, denoted as $O$.
• Standard Tools: A standard grid system (e.g., $1\,cm$ squares) known as graph paper is used for accurate representation.
• Extension of Directions: A full coordinate system includes both positive and negative directions for both axes.
• Mathematical Classification of Lines:
  • Vertical Lines: Exist in the form $x = \text{constant}$. They maintain a fixed distance from the vertical axis and are perpendicular to the horizontal axis.
  • Horizontal Lines: Exist in the form $y = \text{constant}$. They are parallel to the horizontal axis and perpendicular to the vertical axis.

3.1.6 The Cartesian Plane

• Definition: A two-dimensional coordinate system formed by two perpendicular axes.
  • x-axis: The horizontal number line.
  • y-axis: The vertical number line.
• The Origin: The intersection of the two axes, denoted as $O(0, 0)$.
• Quadrants: The axes divide the surface into four distinct regions.
• Interpreting Coordinates:
  • Starting from $O(0, 0)$, the first number signals movement along the $x$-axis (left/negative or right/positive).
  • The second number signals movement along the $y$-axis (down/negative or up/positive).
• Cartesian Coordinates Nomenclature:
  • x-coordinate: Represents horizontal distance from the origin.
  • y-coordinate: Represents vertical distance from the origin.
  • Example: In the pair $(5, 7)$, $5$ is the $x$-coordinate and $7$ is the $y$-coordinate.

3.1.7 Systematic Procedure for Plotting Points

• The Plotting Process:
  1. Begin at the origin ($O$).
  2. Navigate along the $x$-axis according to the amount and direction indicated by the $x$-coordinate.
  3. Move vertically (up or down) parallel to the $y$-axis according to the $y$-coordinate.
  4. Mark the final position with a dot ($.$) or a cross ($\times$).
• Selecting an Appropriate Scale:
  • Evaluate the highest and lowest values in the data set for both $x$ and $y$.
  • The scale should encompass all values while remaining large enough for clarity and precision.
  • Standard Increment Scale: $1\,cm$ usually represents $1, 2, 5, 10, 20, 50, \text{ or } 100$ units.
• Example 3.1: Plotting $(2, -3)$ and $(-2.4, 1.8)$:
  • For $(2, -3)$: Navigate $2$ steps positively on the $x$-axis, then $3$ steps negatively on the $y$-axis.
  • For $(-2.4, 1.8)$: Navigate $2.4$ steps left on the $x$-axis, then $1.8$ steps up parallel to the $y$-axis.
• Example 3.2: Quadrilateral Vertices:
  • Points: $A(-7.5, -5)$, $B(0, -5)$, $C(7.5, 7.5)$, $D(0, 7.5)$.
  • Scaling Insight: A scale of $2\,cm$ representing $5$ units was deemed appropriate.
  • Result: Joining these points forms a parallelogram, with diagonal intersection at $(0, 1.25)$.

3.1.8 - 3.1.9 Linear Graphs and Tables of Values

• Linear Graph Property: This is a graph whose equation represents a straight line. They are essential for modeling relationships with a constant rate of change.
• Linear Equation definition: An equation where the highest power of any variable is $1$.
• Variable Dynamics:
  • Independent Variable (x): The input, which is chosen freely.
  • Dependent Variable (y): The output, determined by the rule of the equation.
• Using a Table of Values:
  • A systematic method to generate ordered pairs by substituting selected $x$ values into the equation.
  • Reliability Rule: While two points uniquely determine a line, it is practical to plot three points. If all three align, the calculations are confirmed as accurate.
• Analysis of $y = 2x + 3$:
  • The coefficient of $x$ ($2$) dictates the slope: as $x$ increases by $1$, $y$ increases by $2$.
  • The constant ($3$) identifies where the line crosses the $y$-axis.
  • Table values: $(0, 3)$, $(1, 5)$, $(2, 7)$, $(3, 9)$, $(4, 11)$.

3.1.10 - 3.1.11 Intercepts: X and Y

• Oblique Lines: Straight lines that are neither horizontal nor vertical. These always intersect both axes exactly once.
• The Y-Intercept:
  • The point where the graph crosses the vertical axis.
  • Condition: $x = 0$.
• The X-Intercept:
  • The point where the graph crosses the horizontal axis.
  • Condition: $y = 0$.
• Algebraic Determination:
  • Example: $3x + 2y = 4$
    • In slope-intercept form ($y = -\frac{3}{2}x + 2$), the $y$-intercept is $(0, 2)$.
    • Setting $y=0$: $3x + 2(0) = 4 \Rightarrow 3x = 4 \Rightarrow x = \frac{4}{3}$. The $x$-intercept is $(\frac{4}{3}, 0)$.
  • Example: $x - y = 3$
    • Setting $x=0$: $y = -3$. The $y$-intercept is $(0, -3)$.
    • Setting $y=0$: $x = 3$. The $x$-intercept is $(3, 0)$.

3.1.11 - 3.1.12 The Gradient (Slope) of a Straight Line

• Conceptual Steepness: Gradient is the measure of how steep a line is, calculated as the ratio of vertical change to horizontal change.
• The Ladder Analogy:
  • Moving vertically ($BC = 2.8\,m$) vs horizontally ($AB = 2\,m$).
  • $\text{Gradient} = \frac{\text{Vertical Distance}}{\text{Horizontal Distance}} = \frac{2.8}{2} = 1.4$.
  • A second position ($BD = 2.5\,m$) yields $\text{Gradient} = \frac{2.8}{2.5} = 1.12$. Because $1.12 < 1.4$, this position is less steep.
• General Formula for Gradient ($m$):
  • Given points $P(x_1, y_1)$ and $Q(x_2, y_2)$, the gradient is:
  • $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Gradient Directions:
  • Positive Gradient: Increasing $x$ results in increasing $y$; the line rises from left to right.
  • Negative Gradient: Increasing $x$ results in decreasing $y$; the line falls from left to right.
• Example 3.3 Calculation:
  • Segment $AB$ involving $A(4, 0)$ and $B(6, 4)$: $m = \frac{4 - 0}{6 - 4} = \frac{4}{2} = 2$.
  • Segment $AC$ involving $A(4, 0)$ and $C(0, 6)$: $m = \frac{6 - 0}{0 - 4} = \frac{6}{-4} = -1.5$.

3.1.13 Forming the Equation of a Line Using Two Points

• Step-by-Step Procedure:
  1. Calculate the Gradient ($m$): Use $m = \frac{y_2 - y_1}{x_2 - x_1}$.
  2. Apply Point-Slope Form: Utilize the formula $y - y_1 = m(x - x_1)$.
  3. Simplify: Rearrange into Slope-Intercept form ($y = mx + c$) or Standard form ($ax + by + c = 0$).
• Demonstration Examples:
  • Line through $(2, 3)$ and $(6, 11)$:
    • $m = \frac{11 - 3}{6 - 2} = 2$.
    • $y - 3 = 2(x - 2) \rightarrow y - 3 = 2x - 4 \rightarrow y = 2x - 1$.
  • Line through $(-1, 4)$ and $(3, -2)$:
    • $m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -1.5$.
    • $y - 4 = -1.5(x + 1) \rightarrow y = -1.5x + 2.5$.

3.1.14 Fundamentals of Linear Equations

• Terminology:
  • Equation: A mathematical sentence utilizing the symbol $=$ to express equality.
  • Conditional Equation: Satisfied by specific values of the unknown (e.g., $x + 11 = 15$, true only if $x = 4$).
  • Identity: An equation that remains true regardless of the values substituted for the unknown variable (e.g., $2x + 2 = 2(x + 1)$).
  • Linear Equation: An equation where the unknown has a power of $1$.
  • Solution: The specific value of the unknown that makes the equation true.
• Algebraic Conventions:
  • Early alphabet letters ($a, b, c$) represent constant values.
  • Late alphabet letters ($x, y, z$) represent unknown values to be solved.

Solving Linear Equations: Methods

• The Balancing Method: Treat the equation like a scale. To maintain balance, perform identical operations on both sides:
  • Add/Subtract the same number.
  • Multiply/Divide by the same non-zero number.
• Example 3.9 Balancing Process for $8x - 6 = 5x + 9$:
  1. Add $6$: $8x = 5x + 15$.
  2. Subtract $5x$: $3x = 15$.
  3. Divide by $3$: $x = 5$.
• The Cover-up Technique: Useful for simple equations. Example: In $2x + 3 = 17$, if $\text{something} + 3 = 17$, then the "something" ($2x$) must be $14$. If $2x = 14$, $x$ must be $7$.
• Dealing with Fractions: Algebraic fractions (e.g., $\frac{x - 4}{3}$) are solved by multiplying every term by the LCM (Least Common Multiple) of all denominators to "clear" the fractions.
  • Example 3.12: $\frac{2y}{5} + \frac{3}{4} = 10 + \frac{y}{2}$.
  • LCM of $5, 4, 2 = 20$.
  • $20(\frac{2y}{5}) + 20(\frac{3}{4}) = 20(10) + 20(\frac{y}{2})$.
  • $8y + 15 = 200 + 10y \rightarrow -2y = 185 \rightarrow y = -92.5$.
• Equations involving Brackets:
  • Brackets express multiplication across quantities: $3(x + 4) = 3x + 12$.
  • Rule: Multiply the factor outside by each term inside. Pay careful attention to signs (e.g., $-a(a + b) = -a^2 - ab$).

Forming and Solving Linear Equations (Word Problems)

• Translation Table (Keywords to Signs):
  • Sum, total, altogether: Add ($+$)
  • Difference, less than: Subtract ($-$)
  • Twice a number: $2x$
  • Shared equally: Divide ($\div$)
  • Is equal to: $=$
• General Steps:
  1. Carefully identify the unknown quantity.
  2. Let a variable (usually $x$) represent the unknown.
  3. Translate the verbal statements into a linear equation.
  4. Solve using balancing/algebraic methods.
  5. Verify the answer against the original problem context.
• Tom and Mary Example (FRW 450 total):
  • Mary gets $54\text{ FRW}$ less than Tom.
  • Let Tom = $x$; Mary = $x - 54$.
  • $x + (x - 54) = 450 \Rightarrow 2x = 504 \Rightarrow x = 252$.
  • Tom receives $252\text{ FRW}$; Mary receives $198\text{ FRW}$.

3.3 Inequalities

• Definition: Statements expressing comparisons such as "greater than" or "fewer than," using symbols rather than an equal sign.
• Core Symbols:
  • $>$: Greater than
  • $<$: Less than
  • $\ge$: Greater than or equal to
  • $\le$: Less than or equal to
• Representation on a Number Line:
  • Open Circle (o): Indicates the endpoint is not included (used for $>$ and $<$).
  • Closed Dot (•): Indicates the endpoint is included (used for $\ge$ and $\le$).
• Solving Rules:
  • Add or subtract the same number from both sides.
  • Multiply or divide by a positive number.
  • Critical Rule: If multiplying or dividing by a negative number, you must reverse the inequality sign.
• Compound Statements: Express that a variable lies between two values, e.g., $a < x < b$
• Word Problem Example (Mangoes):
  • Lucy had $m$ mangoes. Gave away $4$. Half the remainder is less than $17$.
  • $\frac{m - 4}{2} < 17 \Rightarrow m - 4 < 34 \Rightarrow m < 38$.
  • Maximum whole number of mangoes is $37$.
• Word Problem Example (Bus Capacity):
  • Bus carries at most $60$ passengers. $18$ are already on board.
  • $18 + p \le 60 \Rightarrow p \le 42$. Maximum additional boarders = $42$.