6.2 Forms of Linear Equations & Graphing

Overview of Linear Equation Representations

A linear equation in two variables represents a set of ordered pairs $(x, y)$ that form a straight line when plotted on the Cartesian coordinate plane. Every linear equation has degree $1$, meaning the variables $x$ and $y$ appear only to the first power and are never multiplied together, enclosed in roots, or positioned in denominators.

Depending on the given information or the analytical goal, linear equations are expressed in three primary standard forms:

1. Slope-Intercept Form: $y = mx + b$
2. Point-Slope Form: $y - y_1 = m(x - x_1)$
3. Standard Form: $Ax + By = C$

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1. Slope-Intercept Form: $y = mx + b$

Slope-intercept form is the most widely utilized format for graphing and algebraic analysis because both key geometric parameters are explicitly isolated.

$y = mx + b$

• $m$ represents the slope: The rate of vertical displacement per unit horizontal displacement ($\frac{\Delta y}{\Delta x}$).
• $b$ represents the $y$-intercept: The vertical coordinate where the line crosses the $y$-axis, corresponding to the ordered pair $(0, b)$.

Step-by-Step Graphing via Slope-Intercept Form

1. Plot the $y$-intercept: Locate the point $(0, b)$ on the vertical $y$-axis.
2. Use the slope to find a second point: Write $m$ as a fraction $\frac{\text{rise}}{\text{run}}$. From $(0, b)$, move vertically by the rise (up if positive, down if negative) and horizontally by the run (right if positive, left if negative) to plot a second coordinate point $(x_2, y_2)$.
3. Draw the line: Connect the points with a straightedge and extend the line indefinitely with arrows at both ends.

Graphing y = (2/3)x - 1:
  1. Plot y-intercept at (0, -1)
  2. From (0, -1), Rise = +2 (up 2) and Run = +3 (right 3) -> Point (3, 1)
  3. From (3, 1), Rise = +2, Run = +3 -> Point (6, 3)
  4. Draw line through (0, -1), (3, 1), (6, 3)

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2. Point-Slope Form: $y - y_1 = m(x - x_1)$

Point-slope form is derived directly from the fundamental definition of slope. If $(x_1, y_1)$ is a known fixed point on a line and $(x, y)$ is any arbitrary point on the same line, then: $m = \frac{y - y_1}{x - x_1} \implies y - y_1 = m(x - x_1)$

When to Use Point-Slope Form

• Case A: Given a slope $m$ and one coordinate point $(x_1, y_1)$: Substitute $m, x_1,$ and $y_1$ directly into the equation. Example: Line through $(4, -5)$ with slope $m = -\frac{2}{3}$: $y - (-5) = -\frac{2}{3}(x - 4) \implies y + 5 = -\frac{2}{3}(x - 4)$

• Case B: Given two coordinate points $(x_1, y_1)$ and $(x_2, y_2)$:

  1. Calculate the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$.
  2. Choose either point as $(x_1, y_1)$ and write the point-slope equation.
  3. Distribute $m$ and isolate $y$ to convert to slope-intercept form ($y = mx + b$) or rearrange into standard form ($Ax + By = C$).

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3. Standard Form: $Ax + By = C$

In standard algebraic convention on the ACCUPLACER exam, a linear equation in Standard Form satisfies the following rigorous formatting criteria:

$Ax + By = C$

1. $A, B,$ and $C$ must be integers (no fractions or decimals).
2. $A$ and $B$ cannot both be zero ($A^2 + B^2 \neq 0$).
3. $A$ is conventionally non-negative ($A \ge 0$). If $A < 0$, multiply the entire equation by $-1$.
4. Greatest Common Divisor condition: $\gcd(|A|, |B|, |C|) = 1$ (all common integer factors must be cleared).

The Intercept Method of Graphing

Standard form is uniquely suited for rapid graphing via the intercept method:

• Find the $x$-intercept: Set $y = 0$ and solve for $x$: $Ax + B(0) = C \implies Ax = C \implies x = \frac{C}{A} \implies \left(\frac{C}{A}, 0\right)$
• Find the $y$-intercept: Set $x = 0$ and solve for $y$: $A(0) + By = C \implies By = C \implies y = \frac{C}{B} \implies \left(0, \frac{C}{B}\right)$
• Slope Shortcut from Standard Form: Isolating $y$ demonstrates that: $By = -Ax + C \implies y = -\frac{A}{B}x + \frac{C}{B} \implies \mathbf{m = -\frac{A}{B}}$

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Comparison and Conversion Between Linear Forms

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Special Lines: Horizontal and Vertical

Horizontal Line: y = c                     Vertical Line: x = k
           y                                          y
           │                                          │       │
───────────┼─────────── y = c                         │       │ x = k
           │                                          │       │
───────────┼─────────── x                  ───────────┼───────┼─── x
           │                                          │       │ (k,0)
           │                                          │       │
 Slope m = 0, y-int (0,c)                   Slope m = undefined, x-int (k,0)
 Function of x: YES                         Function of x: NO (Fails VLT)

1. Horizontal Lines ($y = c$):

   • Every point on the line shares the identical $y$-coordinate $c$.
   • Slope is $0$ ($m = 0$). In slope-intercept form: $y = 0x + c \implies y = c$.
   • Graph is parallel to the $x$-axis.

2. Vertical Lines ($x = k$):

   • Every point on the line shares the identical $x$-coordinate $k$.
   • Slope is undefined (division by zero: $\Delta x = 0$).
   • Cannot be written in slope-intercept form because $m$ does not exist.
   • Graph is parallel to the $y$-axis and perpendicular to the $x$-axis.
   • Function Warning: A vertical line fails the Vertical Line Test (VLT) and does not define $y$ as a function of $x$.

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Step-by-Step Worked Examples

Worked Example 1: Writing Equations from Two Points in All Three Forms

A line passes through the coordinate points $P(-6, 11)$ and $Q(4, -4)$. Write the equation of the line in (a) Point-Slope Form, (b) Slope-Intercept Form, and (c) Standard Form.

Step 1: Calculate the slope $m$ $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 11}{4 - (-6)} = \frac{-15}{4 + 6} = \frac{-15}{10} = -\frac{3}{2}$

Step 2: Write Point-Slope Form Using point $P(-6, 11)$: $y - 11 = -\frac{3}{2}(x - (-6)) \implies \mathbf{y - 11 = -\frac{3}{2}(x + 6)}$ (Alternatively, using point $Q(4, -4)$: $y + 4 = -\frac{3}{2}(x - 4)$).

Step 3: Convert to Slope-Intercept Form ($y = mx + b$) Distribute the slope across the parentheses: $y - 11 = -\frac{3}{2}x - \frac{3}{2}(6)$ $y - 11 = -\frac{3}{2}x - 9$ Add $11$ to both sides: $\mathbf{y = -\frac{3}{2}x + 2}$ Note: The $y$-intercept is $(0, 2)$.

Step 4: Convert to Standard Form ($Ax + By = C$) Start from slope-intercept form and eliminate fractions by multiplying all terms by the LCD ($2$): $2(y) = 2\left(-\frac{3}{2}x + 2\right)$ $2y = -3x + 4$ Add $3x$ to both sides to bring all variable terms to the left: $\mathbf{3x + 2y = 4}$ Verification: $A = 3 > 0$, $B = 2$, $C = 4$ are all integers with $\gcd(3, 2, 4) = 1$.

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Worked Example 2: Finding Parallel and Perpendicular Lines

Given the reference line $4x - 6y = 15$, find the equation in slope-intercept form for:

1. Line $L_1$ that passes through $(3, -2)$ and is parallel to the reference line.
2. Line $L_2$ that passes through $(3, -2)$ and is perpendicular to the reference line.

Step 1: Determine the slope of the reference line Convert $4x - 6y = 15$ to slope-intercept form: $-6y = -4x + 15 \implies y = \frac{-4}{-6}x + \frac{15}{-6} \implies y = \frac{2}{3}x - \frac{5}{2}$ The reference slope is $m_{\text{ref}} = \frac{2}{3}$.

Step 2: Construct the parallel line $L_1$

• For parallel lines: $m_1 = m_{\text{ref}} = \frac{2}{3}$.
• Apply point-slope form with $(x_1, y_1) = (3, -2)$: $y - (-2) = \frac{2}{3}(x - 3)$ $y + 2 = \frac{2}{3}x - 2$ $y = \frac{2}{3}x - 4$

Step 3: Construct the perpendicular line $L_2$

• For perpendicular lines: $m_2 = -\frac{1}{m_{\text{ref}}} = -\frac{3}{2}$.
• Apply point-slope form with $(x_1, y_1) = (3, -2)$: $y - (-2) = -\frac{3}{2}(x - 3)$ $y + 2 = -\frac{3}{2}x + \frac{9}{2}$ $y = -\frac{3}{2}x + \frac{9}{2} - \frac{4}{2}$ $y = -\frac{3}{2}x + \frac{5}{2}$

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Common Pitfalls & ACCUPLACER Exam Traps

1. Sign Errors When Inserting Negative Coordinates into Point-Slope Form: Writing $y - y_1 = m(x - x_1)$ for point $(-4, 7)$ as $y - 7 = m(x - 4)$ is a fatal error. The correct expression is $x - (-4) = x + 4$.
2. Violating Standard Form Integer and Positivity Conventions: Leaving an equation as $\frac{1}{2}x - \frac{3}{4}y = 5$ or $-4x + 6y = -10$ violates standard form. Multiply by LCDs and $-1$ to obtain proper integer coefficients with $A > 0$.
3. Confusing Horizontal and Vertical Line Equations: Students frequently write $x = 4$ for a horizontal line crossing $(0, 4)$. Remember: horizontal lines set $y$ equal to a constant ($y = 4$), while vertical lines set $x$ equal to a constant ($x = 4$).
4. Incorrect Slope from Standard Form: Assuming the slope of $Ax + By = C$ is $\frac{A}{B}$ rather than $-\frac{A}{B}$. In $5x + 2y = 10$, the slope is $m = -\frac{5}{2}$, NOT $+\frac{5}{2}$.