Comprehensive Guide to the Coordinate Plane, Slope, and Linear Equations

Foundations of the Coordinate Plane

• Structure of the Coordinate System:

  • The coordinate plane is formed by the intersection of two perpendicular number lines.
  • The x-axis: The horizontal number line on the coordinate plane.
  • The y-axis: The vertical number line on the coordinate plane.
  • The Origin: The point where the x-axis and y-axis intersect, defined by the coordinates $(0, 0)$.
  • Ordered Pairs: Points on the coordinate plane are identified using ordered pairs written in the standard format $(x, y)$, where $x$ represents the horizontal directed distance from the y-axis and $y$ represents the vertical directed distance from the x-axis.

• Quadrants and Coordinate Signs:

  • The axes divide the plane into four distinct regions called quadrants, numbered counterclockwise from I to IV:
  • Quadrant I: Both coordinates are positive ($x > 0$, $y > 0$).
  • Quadrant II: The x-coordinate is negative and the y-coordinate is positive ($x < 0$, $y > 0$). For example, the point $(-8, 5)$ lies in Quadrant II.
  • Quadrant III: Both coordinates are negative ($x < 0$, $y < 0$).
  • Quadrant IV: The x-coordinate is positive and the y-coordinate is negative ($x > 0$, $y < 0$).
  • Points on the Axes: Any point with a zero coordinate lies directly on an axis and does not belong to any quadrant. For example, the point $(0, 4)$ lies on the y-axis and therefore lies in no quadrant.

• Identifying and Plotting Points on a Graph:

  • Coordinates must always be enclosed in parentheses and separated by a comma: $(x, y)$.
  • Coordinates corresponding to labeled points on a standard grid:
  • Point $R$ is located at $(2, 2)$
  • Point $S$ is located at $(-2, 4)$
  • Point $T$ is located at $(-3, 2)$
  • Point $U$ is located at $(3, -2)$
  • Point $V$ is located at $(-2, -3)$
  • Point $W$ is located at $(0, -2)$
  • Point $X$ is located at $(2, -3)$
  • Point $Y$ is located at $(3, 0)$
  • Point $Z$ is located at $(4, 3)$
  • Point $Q$ is located at $(-1, 0)$
  • Graph points identified by given coordinate pairs:
  • Coordinates $(-5, 1)$ identify point $G$
  • Coordinates $(-2, -4)$ identify point $K$
  • Coordinates $(0, 5)$ identify point $B$
  • Coordinates $(5, 3)$ identify point $E$
  • Coordinates $(3, -3)$ identify point $P$
  • Coordinates $(-4, -3)$ identify point $L$
  • Coordinates $(2, 5)$ identify point $D$
  • Coordinates $(5, 0)$ identify point $F$
  • Coordinates $(0, -3)$ identify point $N$
  • Coordinates $(-4, 5)$ identify point $A$

Evaluating Linear Equations and Verifying Solutions

• Evaluating Linear Equations for a Specific Value ($x = -6$):

  • Equation $y = -2x + 1$:
  • Substitute $x = -6$: $y = -2(-6) + 1 = 12 + 1 = 13$
  • Solution ordered pair: $(-6, 13)$
  • Equation $x + y = -2$:
  • Substitute $x = -6$: $-6 + y = -2$
  • Add $6$ to both sides: $y = 4$
  • Solution ordered pair: $(-6, 4)$
  • Equation $5x - 5 = y$:
  • Substitute $x = -6$: $5(-6) - 5 = y$
  • Evaluate: $-30 - 5 = -35$
  • Solution ordered pair: $(-6, -35)$

• Testing Ordered Pairs as Solutions to $y = -3x - 4$:

  • Test point $(1, 7)$:
  • Substitute into equation: $7 = -3(1) - 4$
  • Evaluate right side: $7 = -3 - 4 = -7$
  • Result: $7 \neq -7$. The point is not a solution.

• Test point $(2, -9)$:

  • Substitute into equation: $-9 = -3(2) - 4$
  • Evaluate right side: $-9 = -6 - 4 = -10$
  • Result: $-9 \neq -10$. The point is not a solution.
  • Test point $(-3, 5)$:
  • Substitute into equation: $5 = -3(-3) - 4$
  • Evaluate right side: $5 = 9 - 4 = 5$
  • Result: $5 = 5$. The point is a solution.

• Testing Ordered Pairs as Solutions to $4x - 3y = 6$:

  • Test point $(3, 2)$:
  • Substitute into equation: $4(3) - 3(2) = 6$
  • Evaluate left side: $12 - 6 = 6$
  • Result: $6 = 6$. The point is a solution.
  • Test point $(-3, -2)$:
  • Substitute into equation: $4(-3) - 3(-2) = 6$
  • Evaluate left side: $-12 + 6 = -6$
  • Result: $-6 \neq 6$. The point is not a solution.

• Test point $(0, 2)$:

  • Substitute into equation: $4(0) - 3(2) = 6$
  • Evaluate left side: $0 - 6 = -6$
  • Result: $-6 \neq 6$. The point is not a solution.

• Verifying Points on Specific Linear Equations:

  • Is the point $(-2, -3)$ a solution to $y + 5 = -x$?
  • Substitute $x = -2$ and $y = -3$:
  • Left side: $-3 + 5 = 2$
  • Right side: $-(-2) = 2$
  • Conclusion: $2 = 2$. Yes, the point is a solution.
  • Is the point $(-2, -9)$ on the line $y - 2x = 5$?
  • Substitute $x = -2$ and $y = -9$:
  • Left side: $-9 - 2(-2) = -9 + 4 = -5$
  • Right side: $5$
  • Conclusion: $-5 \neq 5$. No, the point is not on the line.

• Is the point $(3, -2)$ on the line $y = \frac{1}{3}x - 3$?

  • Substitute $x = 3$ and $y = -2$:
  • Right side: $\frac{1}{3}(3) - 3 = 1 - 3 = -2$
  • Left side: $-2$
  • Conclusion: $-2 = -2$. Yes, the point is on the line.

• Is the point $(4, 6)$ a solution to $y = -\frac{1}{2}x + 4$?

  • Substitute $x = 4$ and $y = 6$:
  • Right side: $-\frac{1}{2}(4) + 4 = -2 + 4 = 2$
  • Left side: $6$
  • Conclusion: $6 \neq 2$. No, the point is not a solution.

• Algebraic Equation Solving and Verification Review:

  • Equation 1: $-7 + 9y + 2 - 3y = -29$
  • Combine like terms on the left: $6y - 5 = -29$
  • Add $5$ to both sides: $6y = -24$
  • Divide by $6$: $y = -4$
  • Check substitution: $-7 + 9(-4) + 2 - 3(-4) = -7 - 36 + 2 + 12 = -43 + 14 = -29$. Confirmed.
  • Equation 2: $4(d + 8) = 6(d - 6)$
  • Apply distributive property: $4d + 32 = 6d - 36$
  • Subtract $32$ from both sides: $4d = 6d - 68$
  • Subtract $6d$ from both sides: $-2d = -68$
  • Divide by $-2$: $d = 34$
  • Check substitution:
    • Left side: $4(34 + 8) = 4(42) = 168$
    • Right side: $6(34 - 6) = 6(28) = 168$
    • Confirmed: $168 = 168$.

Graphing Linear Equations via Tables of Values

• Table Construction Strategy:

  • When graphing linear equations with fractions, choose values of $x$ that are multiples of the denominator to yield integer values for $y$.
  • Standard protocol requires evaluating five domain points: two positive numbers, two negative numbers, and zero.

• Equation 1: $y = \frac{1}{4}x - 2$

  • Multiples of $4$ selected for $x$:
  • For $x = 8$: $y = \frac{1}{4}(8) - 2 = 2 - 2 = 0 \rightarrow (8, 0)$
  • For $x = 4$: $y = \frac{1}{4}(4) - 2 = 1 - 2 = -1 \rightarrow (4, -1)$
  • For $x = 0$: $y = \frac{1}{4}(0) - 2 = 0 - 2 = -2 \rightarrow (0, -2)$
  • For $x = -4$: $y = \frac{1}{4}(-4) - 2 = -1 - 2 = -3 \rightarrow (-4, -3)$
  • For $x = -8$: $y = \frac{1}{4}(-8) - 2 = -2 - 2 = -4 \rightarrow (-8, -4)$

• Equation 2: $y = -\frac{1}{2}x + 5$

  • Multiples of $2$ selected for $x$:
  • For $x = 4$: $y = -\frac{1}{2}(4) + 5 = -2 + 5 = 3 \rightarrow (4, 3)$
  • For $x = 2$: $y = -\frac{1}{2}(2) + 5 = -1 + 5 = 4 \rightarrow (2, 4)$
  • For $x = 0$: $y = -\frac{1}{2}(0) + 5 = 0 + 5 = 5 \rightarrow (0, 5)$
  • For $x = -2$: $y = -\frac{1}{2}(-2) + 5 = 1 + 5 = 6 \rightarrow (-2, 6)$
  • For $x = -4$: $y = -\frac{1}{2}(-4) + 5 = 2 + 5 = 7 \rightarrow (-4, 7)$

• Equation 3: $y = \frac{2}{3}x - 3$

  • Multiples of $3$ selected for $x$:
  • For $x = 6$: $y = \frac{2}{3}(6) - 3 = 4 - 3 = 1 \rightarrow (6, 1)$
  • For $x = 3$: $y = \frac{2}{3}(3) - 3 = 2 - 3 = -1 \rightarrow (3, -1)$
  • For $x = 0$: $y = \frac{2}{3}(0) - 3 = 0 - 3 = -3 \rightarrow (0, -3)$
  • For $x = -3$: $y = \frac{2}{3}(-3) - 3 = -2 - 3 = -5 \rightarrow (-3, -5)$
  • For $x = -6$: $y = \frac{2}{3}(-6) - 3 = -4 - 3 = -7 \rightarrow (-6, -7)$

• Equation 4: $y = \frac{2}{5}x + 1$

  • Multiples of $5$ selected for $x$:
  • For $x = 10$: $y = \frac{2}{5}(10) + 1 = 4 + 1 = 5 \rightarrow (10, 5)$
  • For $x = 5$: $y = \frac{2}{5}(5) + 1 = 2 + 1 = 3 \rightarrow (5, 3)$
  • For $x = 0$: $y = \frac{2}{5}(0) + 1 = 0 + 1 = 1 \rightarrow (0, 1)$
  • For $x = -5$: $y = \frac{2}{5}(-5) + 1 = -2 + 1 = -1 \rightarrow (-5, -1)$
  • For $x = -10$: $y = \frac{2}{5}(-10) + 1 = -4 + 1 = -3 \rightarrow (-10, -3)$

• Equation 5: $y = -\frac{1}{4}x + 5$

  • Multiples of $4$ selected for $x$:
  • For $x = 8$: $y = -\frac{1}{4}(8) + 5 = -2 + 5 = 3 \rightarrow (8, 3)$
  • For $x = 4$: $y = -\frac{1}{4}(4) + 5 = -1 + 5 = 4 \rightarrow (4, 4)$
  • For $x = 0$: $y = -\frac{1}{4}(0) + 5 = 0 + 5 = 5 \rightarrow (0, 5)$
  • For $x = -4$: $y = -\frac{1}{4}(-4) + 5 = 1 + 5 = 6 \rightarrow (-4, 6)$
  • For $x = -8$: $y = -\frac{1}{4}(-8) + 5 = 2 + 5 = 7 \rightarrow (-8, 7)$

• Equation 6: $y = \frac{1}{3}x + 2$

  • Multiples of $3$ selected for $x$:
  • For $x = 6$: $y = \frac{1}{3}(6) + 2 = 2 + 2 = 4 \rightarrow (6, 4)$
  • For $x = 3$: $y = \frac{1}{3}(3) + 2 = 1 + 2 = 3 \rightarrow (3, 3)$
  • For $x = 0$: $y = \frac{1}{3}(0) + 2 = 0 + 2 = 2 \rightarrow (0, 2)$
  • For $x = -3$: $y = \frac{1}{3}(-3) + 2 = -1 + 2 = 1 \rightarrow (-3, 1)$
  • For $x = -6$: $y = \frac{1}{3}(-6) + 2 = -2 + 2 = 0 \rightarrow (-6, 0)$

Concept and Calculation of Slope

• Definition and Directionality of Slope:

  • The term slope describes the steepness or slant of a straight line.
  • When reading a graph from left to right:
  • A line that slants upward has a positive slope ($m > 0$).
  • A line that slants downward has a negative slope ($m < 0$).

• Mathematical Formulations of Slope:

  • Slope is defined as the ratio of vertical change to horizontal change, commonly expressed as "rise over run":
  • $\text{Slope } m = \frac{\text{Rise}}{\text{Run}}$
  • Rise ($\boldsymbol{\Delta y}$): Vertical change or difference between the y-values of any two points on the line.
  • Run ($\boldsymbol{\Delta x}$): Horizontal change or difference between the x-values of the same two points on the line.
  • The Slope Formula:
  • Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope is calculated as:       $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

• Direct Graphical Calculations of Slope:

  • Example with positive slope: A graph exhibiting a vertical change of $3$ units upward for every $1$ unit rightward:
  • $\text{Rise } \boldsymbol{\Delta y} = 3$, $\text{Run } \boldsymbol{\Delta x} = 1$
  • $m = \frac{3}{1} = 3$
  • Example with negative slope: A graph exhibiting a vertical change of $3$ units downward for every $2$ units rightward:
  • $\text{Rise } \boldsymbol{\Delta y} = -3$, $\text{Run } \boldsymbol{\Delta x} = 2$
  • $m = -\frac{3}{2}$

• Two-Point Slope Formula Calculations:

  • Given points $(6, 4)$ and $(-3, 5)$:
  • Assign $(x_1, y_1) = (6, 4)$ and $(x_2, y_2) = (-3, 5)$
  • $m = \frac{5 - 4}{-3 - 6} = \frac{1}{-9} = -\frac{1}{9}$
  • Given points $(1, 2)$ and $(-4, 3)$:
  • Assign $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (-4, 3)$
  • $m = \frac{3 - 2}{-4 - 1} = \frac{1}{-5} = -\frac{1}{5}$
  • Given points $(-3, 1)$ and $(2, 7)$:
  • Assign $(x_1, y_1) = (-3, 1)$ and $(x_2, y_2) = (2, 7)$
  • $m = \frac{7 - 1}{2 - (-3)} = \frac{6}{2 + 3} = \frac{6}{5}$
  • Given points $(15, 5)$ and $(3, 1)$:
  • Assign $(x_1, y_1) = (15, 5)$ and $(x_2, y_2) = (3, 1)$
  • $m = \frac{1 - 5}{3 - 15} = \frac{-4}{-12} = \frac{1}{3}$
  • Given points $(-6, 5)$ and $(3, -3)$:
  • Assign $(x_1, y_1) = (-6, 5)$ and $(x_2, y_2) = (3, -3)$
  • $m = \frac{-3 - 5}{3 - (-6)} = \frac{-8}{3 + 6} = -\frac{8}{9}$

• Special Cases of Slope:

  • Horizontal Lines (Zero Slope):
  • For points $(-3, 2)$ and $(1, 2)$:
    • Difference in y: $\boldsymbol{\Delta y} = 2 - 2 = 0$
    • Difference in x: $\boldsymbol{\Delta x} = 1 - (-3) = 4$
    • $m = \frac{0}{4} = 0$
  • Any completely horizontal line has a slope of zero.
  • Vertical Lines (Undefined Slope):
  • For points $(5, 3)$ and $(5, -2)$:
    • Difference in y: $\boldsymbol{\Delta y} = -2 - 3 = -5$
    • Difference in x: $\boldsymbol{\Delta x} = 5 - 5 = 0$
    • $m = \frac{-5}{0}$
  • Division by zero is mathematically undefined. Therefore, any completely vertical line has an undefined slope.

• Comparative and Physical Applications of Slope:

  • Slope represents the rate of physical steepness or rate of change in practical scenarios:
  • Pushing a heavy cart up an incline: Among slopes of $\frac{1}{2}$, $\frac{5}{6}$, and $3$, a slope of $\frac{1}{2}$ is the easiest because it has the smallest vertical incline per unit of horizontal run.
  • Skiing downhill speed: Among slopes of $-\frac{1}{2}$, $-1$, and $-2$, a slope of $-2$ produces the greatest speed because it represents the steepest descent (greatest absolute rate of vertical drop).
  • Roofer safety: Among roof pitches of $\frac{1}{16}$, $\frac{3}{10}$, and $\frac{3}{2}$, a slope of $\frac{3}{2}$ is the most dangerous for repair work because its steepness exceeds $1$ (a rise of $1.5$ units per $1$ unit of run).

• Finding Slope from Tabular Data:

  • Main School Theater Ticket Sales:
  • Table pairs:
    • $50$ tickets sold: $\$225$
    • $100$ tickets sold: $\$450$
    • $150$ tickets sold: $\$675$
    • $200$ tickets sold: $\$900$
  • Select points $(50, 225)$ and $(100, 450)$
  • $m = \frac{450 - 225}{100 - 50} = \frac{225}{50} = \frac{9}{2} = 4.5$
  • Rate of change: The rate is $\text{\$}4.50$ collected per ticket sold.
  • Calories in Blueberries:
  • Table pairs:
    • $5$ blueberries ($x$): $4$ calories ($y$)
    • $10$ blueberries ($x$): $8$ calories ($y$)
    • $15$ blueberries ($x$): $12$ calories ($y$)
  • Select points $(5, 4)$ and $(10, 8)$
  • $m = \frac{8 - 4}{10 - 5} = \frac{4}{5} = 0.8$
  • Rate of change: The rate is $\frac{4}{5}$ (or $0.8$) calories per blueberry.

Slope-Intercept Form Analysis and Equation Conversion

• Structure and Principles of Slope-Intercept Form:

  • Solving a two-variable linear equation for $y$ places it into slope-intercept form:     $y = mx + b$
  • The Slope ($m$): The coefficient multiplied by $x$, representing the ratio of vertical change to horizontal change ($\frac{\Delta y}{\Delta x}$).
  • The y-intercept ($b$): The constant term, representing the value of $y$ when $x = 0$. This is the point $(0, b)$ where the line intersects the y-axis.

• Identifying Slope and y-intercept from Equations and Graphs:

  • Equation $y = \frac{1}{2}x + 4$:
  • Slope: $m = \frac{1}{2}$
  • y-intercept: $b = 4$
  • Equation $y = -3x - 1$:
  • Slope: $m = -3$
  • y-intercept: $b = -1$
  • Equation $y = -\frac{2}{5}x$:
  • Slope: $m = -\frac{2}{5}$
  • y-intercept: $b = 0$
  • Equation with $m = -\frac{6}{8}$ and $b = -1$:
  • Slope: $m = -\frac{6}{8} = -\frac{3}{4}$
  • y-intercept: $b = -1$
  • Equation: $y = -\frac{6}{8}x - 1$
  • Equation $y = \frac{1}{4}x - 7$:
  • Slope: $m = \frac{1}{4}$
  • y-intercept: $b = -7$
  • Equation with $m = -\frac{1}{4}$ and $b = 3$:
  • Slope: $m = -\frac{1}{4}$
  • y-intercept: $b = 3$
  • Equation: $y = -\frac{1}{4}x + 3$
  • Graph with line having a rise of $-3$ and run of $4$ passing through $(0, 6)$:
  • Slope: $m = -\frac{3}{4}$
  • y-intercept: $b = 6$
  • Equation: $y = -\frac{3}{4}x + 6$

• Rewriting Equations in Slope-Intercept Form ($y = mx + b$):

  • Equation: $3 + y = -3x$
  • Subtract $3$ from both sides: $y = -3x - 3$
  • Slope: $m = -3$
  • y-intercept: $b = -3$
  • Equation: $y - 4x = 6$
  • Add $4x$ to both sides: $y = 4x + 6$
  • Slope: $m = 4$
  • y-intercept: $b = 6$
  • Equation: $y + \frac{1}{2}x = -2$
  • Subtract $\frac{1}{2}x$ from both sides: $y = -\frac{1}{2}x - 2$
  • Slope: $m = -\frac{1}{2}$
  • y-intercept: $b = -2$
  • Equation: $y - \frac{2}{3}x = 9$
  • Add $\frac{2}{3}x$ to both sides: $y = \frac{2}{3}x + 9$
  • Slope: $m = \frac{2}{3}$
  • y-intercept: $b = 9$

• Constructing Equations from Geometric Properties:

  • A line with a y-intercept of $1$ and a slope of $2$:
  • Slope-intercept model: $y = mx + b$
  • Written as: $y = 2x + 1$ (noted alternatively as $y = \frac{1}{2}x + 1$ when evaluated using a slope of $\frac{1}{2}$)
  • A line with a rise of $3$, a run of $-2$, and passing through the origin:
  • Slope: $m = \frac{3}{-2} = -\frac{3}{2}$
  • Passes through the origin $(0, 0)$, so $b = 0$
  • Equation: $y = -\frac{3}{2}x + 0$, or simply $y = -\frac{3}{2}x$

• Deriving Slope and y-intercept from Data Tables:

  • Table 1:
  • $x$-values: $0, 2, 4, 6, 8$
  • $y$-values: $1, 7, 13, 19, 25$
  • Determine slope: $m = \frac{\Delta y}{\Delta x} = \frac{7 - 1}{2 - 0} = \frac{6}{2} = 3$
  • Determine y-intercept: When $x = 0$, $y = 1$, so $b = 1$
  • Equation: $y = 3x + 1$
  • Table 2:
  • $x$-values: $5, 10, 15, 20$
  • $y$-values: $140, 120, 100, 80$
  • Determine slope: $m = \frac{\Delta y}{\Delta x} = \frac{120 - 140}{10 - 5} = \frac{-20}{5} = -4$
  • Determine y-intercept: Extrapolating back to $x = 0$ using $y - mx$ yields $140 - (-4)(5) = 140 + 20 = 160$ (or using base initial value $b = 140$ at step 1)

Graphing Linear Equations Using Slope-Intercept Form

• Four-Step Graphing Procedure:

  • Step 1: Ensure the linear equation is written in slope-intercept form: $y = mx + b$.
  • Step 2: Identify the slope ($m$) and the y-intercept ($b$). Write the slope explicitly as a fraction ($\frac{\text{rise}}{\text{run}}$).
  • Step 3: Plot the y-intercept on the y-axis at the coordinate $(0, b)$.
  • Step 4: From the plotted y-intercept, use the slope ratio ($\frac{\text{rise}}{\text{run}}$) to count vertically and horizontally to plot additional points, then draw a straight line through the points.

• Step-by-Step Graphing Walkthroughs:

  • Graphing $y = -3x - 2$:
  • Slope: $m = -3 = -\frac{3}{1} = \frac{-3}{1} = \frac{3}{-1}$
  • y-intercept: $b = -2$
  • Initial point: Plot $(0, -2)$ on the y-axis.
  • From $(0, -2)$, count down $3$ units and right $1$ unit to plot $(1, -5)$, or count up $3$ units and left $1$ unit to plot $(-1, 1)$.
  • Draw the straight line passing through these points.
  • Graphing $y = \frac{1}{2}x + 4$:
  • Slope: $m = \frac{1}{2}$
  • y-intercept: $b = 4$
  • Initial point: Plot $(0, 4)$ on the y-axis.
  • From $(0, 4)$, count up $1$ unit and right $2$ units to plot $(2, 5)$, or count down $1$ unit and left $2$ units to plot $(-2, 3)$.
  • Draw the straight line passing through these points.
  • Graphing $y = -\frac{2}{3}x + 1$:
  • Slope: $m = -\frac{2}{3} = \frac{-2}{3} = \frac{2}{-3}$
  • y-intercept: $b = 1$
  • Initial point: Plot $(0, 1)$ on the y-axis.
  • From $(0, 1)$, count down $2$ units and right $3$ units to plot $(3, -1)$, or count up $2$ units and left $3$ units to plot $(-3, 3)$.
  • Draw the straight line passing through these points.
  • Graphing $y = x + 4$:
  • Slope: $m = 1 = \frac{1}{1} = \frac{-1}{-1}$
  • y-intercept: $b = 4$
  • Initial point: Plot $(0, 4)$ on the y-axis.
  • From $(0, 4)$, count up $1$ unit and right $1$ unit to plot $(1, 5)$, or count down $1$ unit and left $1$ unit to plot $(-1, 3)$.
  • Draw the straight line passing through these points.