Chapter 3.4 - 3.7

Warm-up Exercises

Exercise 1: Write the Equation of a Line

• Task: Write the equation of the line in slope-intercept form that has a slope of -3 and a y-intercept of 4.

• Solution: The slope-intercept form of a line is given by the equation: $y = mx + b$, where:

  • $m$ is the slope

  • $b$ is the y-intercept.

• For this exercise:

  • Given slope, $m = -3$

  • Given y-intercept, $b = 4$

• The equation thus becomes:
  $y = -3x + 4$

Exercise 2: Rewrite Equation in Slope-Intercept Form

• Task: Write the equation in slope-intercept form:
  $(y + 2) = (x - 5)$

• Steps to Convert:

  1. Distribute the right side: $y + 2 = x - 5$

  2. Isolate $y$: $y = x - 5 - 2$

  3. Final equation: $y = x - 7$

Exercise 3: Graphing the Equation Using Intercepts

• Equation: $-6x + 8y = 24$

• Intercepts Calculation:

  1. X-intercept: Set $y = 0$:

  • $-6x + 8(0) = 24$

  • $-6x = 24$

  • $x = -4$

  1. Y-intercept: Set $x = 0$:

  • $-6(0) + 8y = 24$

  • $8y = 24$

  • $y = 3$

• Intercepts:

  • X-intercept: $(−4,0)$

  • Y-intercept: $(0,3)$

Exercise 4: Farm Problem

• Context: Mr. Jeans raises cows and chickens. Total legs = 140.

• Equation: The function representing total legs: $4x + 2y = 140$

  • Where:

  • $x$ = number of cows

  • $y$ = number of chickens

• Graph the Function:

  • Find the intercepts:

  • X-intercept (Cows): Set $y = 0$:

    • $4x + 2(0) = 140$

    • $4x = 140$

    • $x = 35$

  • Y-intercept (Chickens): Set $x = 0$:

    • $4(0) + 2y = 140$

    • $2y = 140$

    • $y = 70$

• Intercepts:

  • X-intercept: $(35, 0)$

  • Y-intercept: $(0, 70)$

• Interpretation:

  • If there are 0 cows, there can be 70 chickens.

  • If there are 35 cows, there could be 0 chickens.

Course 3, Lesson 3-6: Expressions and Equations

Warm-up

Slope Calculation Using Slope Formula (Do NOT graph)

• Slope Formula:
  m= = (y2 - y1) ÷ (x2 - x1)

Example Calculations

• 1. Points: A(4, 1); B(12, 3)

  • $m = \frac{3 - 1}{12 - 4} = \frac{2}{8} = \frac{1}{4}$

• 2. Points: P(−1, −5); Q(1, -10)

  • $m = \frac{-10 + 5}{1 + 1} = \frac{-5}{2}$

• 3. Points: W(6, 9); Z(12, 9)

  • $m = \frac{9 - 9}{12 - 6} = \frac{0}{6} = 0$

Understanding Forms of Linear Equations

Point-Slope Form and Slope-Intercept Form

• Equation Forms:

  1. Point-Slope Form: (y - y1) = m(x - x1)

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

• Relationship:

  • You can convert between these two forms using algebraic manipulation.

Step-by-Step Examples

Example 1: Point-Slope Form to Slope-Intercept Form

• Given: Write an equation in point-slope form for the line passing through (−2, 3) with slope 4.

• Point-Slope Form: $y - 3 = 4(x + 2)$

  • Rearranging to slope-intercept:
    $y = 4x + 8 - 3$
    $y = 4x + 5$

Example 2: Rewrite Given Equations

• a) $(y - 3 = 4(x + 2))$ to slope-intercept form:
  $y = 4x + 8 - 3$
  $y = 4x + 5$

• b) $(y - 4 = -2(x - 2))$
  $y = -2x + 4 + 4$
  $y = -2x + 8$

Example 3: Constructing Equations from Points

• Task: Write the equation in point-slope form and slope-intercept form for a line that passes through (8, 1) and (−2, 9).

• Steps:

  1. Calculate slope: $m = \frac{9 - 1}{-2 - 8} = \frac{8}{-10} = -\frac{4}{5}$

  2. Using point-slope form:
     $y - 1 = -\frac{4}{5}(x - 8)$

Graphing and Interpretation of Equations

Example: Interpretation of Functions

Cost Interpretation for Dog Training Sessions

• Cost Equation: Let $y$ be cost of attending $x$ sessions.

• Descriptive Table: Utilizes linear forms to represent total cost. Graph this function.

Practical Examples

School Yearbook Sales Example

• Equation: Given total revenue is $60x + 15y = 4740$.

• Interpretation: Find x- and y-intercepts:

  • X-intercept (when y=0): identifies total print yearbooks.

  • Y-intercept (when x=0): identifies total digital yearbooks sold.

• Graph Interpretation: Use as a visual aid in presenting the relationship between number of books sold vs. revenue generated.

Additional Examples & Interpretations

• Drama Ticket Sales Example: Validate graph interpretation with ticket equations to understand revenue breakdown by type of ticket sales using the given equation $5x + 9y = 1260$.

Summary on Linear Equation Analysis

• Concepts Covered:

  • Conversion between forms of equations.

  • Application of equations in real-world settings.

  • Interpretation of x- and y-intercepts for various situations to understand constraints and possibilities within graphing contexts.