Algebra I: Linear Equations, Intercepts, and Real-World Modeling

Graphing Vocabulary and Fundamental Definitions

• Describe: In mathematical contexts, describing a graph requires explaining features clearly using words rather than just providing numeric answers.

• Essential terminology for describing linear graphs:

  • Coordinates

  • Coordinate plane

  • $x$-coordinate

  • $y$-coordinate

  • $x$-intercept

  • $y$-intercept

Formatting, Finding, and Graphing Intercepts

• Coordinates must always be enclosed in parentheses, such as $(x, y)$. Omitting parentheses prevents numbers from being recognized as valid coordinates and results in lost points on assessments.

• $x$-intercept: The point where a line crosses the $x$-axis.

  • General form: $(x, 0)$.

  • Procedure: Substitute $y = 0$ into the equation and solve for $x$.

• $y$-intercept: The point where a line crosses the $y$-axis.

  • General form: $(0, y)$.

  • Procedure: Substitute $x = 0$ into the equation and solve for $y$.

• Example with given intercepts:

  • $x$-intercept: $(3, 0)$

  • $y$-intercept: $(0, -2)$

• Example with equation $2x - 3y = 6$:

  • Finding $x$-intercept: Substitute y = 0 \formatError{ \to } 2x - 3(0) = 6 \formatError{ \to } 2x = 6 \formatError{ \to } x = 3. Coordinates are $(3, 0)$.

  • Finding $y$-intercept: Substitute x = 0 \formatError{ \to } 2(0) - 3y = 6 \formatError{ \to } -3y = 6 \formatError{ \to } y = -2. Coordinates are $(0, -2)$.

• Graphing procedure:

  • Plotting any two distinct points on a line is sufficient to graph the line completely.

  • Use graph paper and a straight edge (such as a ruler or the edge of a note card) to produce precise, straight lines.

  • Verification technique: Substitute any other coordinate point $(x, y)$ from the drawn line back into the original equation; if the point makes the equation true, the graph is verified as correct.

Standard Form and Special Linear Cases

• The standard form of a linear equation is written as $ax + by = c$ (or $Ax + By = C$), where $a$, $b$, and $c$ are real numbers.

• Horizontal Line Case ($a = 0$):

  • Setting $a = 0$ eliminates the $x$-term: 0x + by = c \formatError{ \to } by = c \formatError{ \to } y = \frac{c}{b}.

  • Example: 4y = 12 \formatError{ \to } y = \frac{12}{4} = 3.

  • Slope: A horizontal line is flat across and has a slope of $0$ (no slope).

  • Axis intersections: A horizontal line never crosses the $x$-axis unless $y = 0$, in which case the line is identical to the $x$-axis itself.

• Vertical Line Case ($b = 0$):

  • Setting $b = 0$ eliminates the $y$-term: ax + 0y = c \formatError{ \to } ax = c \formatError{ \to } x = \frac{c}{a}.

  • Slope: A vertical line has an impossible slope, defined as undefined.

  • Axis intersections: A vertical line crosses the $x$-axis, but never crosses the $y$-axis unless $x = 0$, in which case the line is identical to the $y$-axis itself.

• Origin Intersection Case ($c = 0$):

  • Setting $c = 0$ yields $ax + by = 0$.

  • The $x$-intercept is $(0, 0)$ and the $y$-intercept is $(0, 0)$, meaning the line passes directly through the origin.

Modeling Real-World Contexts: Variable Definitions and Restrictions

• Applying linear equations in standard form to real-world scenarios requires identifying the specific meaning of each variable alongside the numerical coefficients.

• General step-by-step procedure for real-world modeling:

  1. Define each variable explicitly with units.

  2. Determine domain and range restrictions based on real-world constraints.

  3. Formulate the linear equation relating quantities, costs, and totals.

  4. Evaluate algebraic solutions against real-world domain and range limits.

Real-World Example 1: Party Punch Ingredients

• Scenario parameters:

  • Lemon-lime soda cost: $1.00\,\text{dollar}$ per bottle.

  • Cranberry-apple juice cost: $1.50\,\text{dollars}$ per bottle.

  • Total expenditure target: $9.00\,\text{dollars}$.

• Variable definitions:

  • Let $x$ = number of bottles of lemon-lime soda purchased.

  • Let $y$ = number of bottles of cranberry-apple juice purchased.

• Domain and range constraints:

  • Retail stores do not sell fractional or partial bottles.

  • $x$ and $y$ cannot be general real numbers; domain and range must be restricted strictly to whole numbers ($\{0, 1, 2, 3, \dots\}$).

• Equation setup:

  • $\text{Cost of soda} + \text{Cost of juice} = \text{Total cost}$

  • 1.00x + 1.50y = 9.00 \formatError{ \to } x + 1.5y = 9

• Point evaluation and testing:

  • Scenario A ($x = 6$): 6 + 1.5y = 9 \formatError{ \to } 1.5y = 3 \formatError{ \to } y = 2. Coordinates: $(6, 2)$. Valid whole-number purchase.

  • Scenario B ($x = 3$): 3 + 1.5y = 9 \formatError{ \to } 1.5y = 6 \formatError{ \to } y = 4. Coordinates: $(3, 4)$. Valid whole-number purchase.

  • Scenario C ($y = 3$): x + 1.5(3) = 9 \formatError{ \to } x + 4.5 = 9 \formatError{ \to } x = 4.5. Coordinates: $(4.5, 3)$. Invalid solution because partial bottles ($4.5$) cannot be bought.

  • Scenario D ($y = 2$): x + 1.5(2) = 9 \formatError{ \to } x + 3 = 9 \formatError{ \to } x = 6. Yields $(6, 2)$, confirming unique functional mapping.

  • Scenario E ($y = 0$): x + 1.5(0) = 9 \formatError{ \to } x = 9. Coordinates: $(9, 0)$. Valid whole-number purchase.

  • Scenario F ($x = 0$): 0 + 1.5y = 9 \formatError{ \to } y = 6. Coordinates: $(0, 6)$. Valid whole-number purchase.

• Graphing discrete real-world solutions:

  • The $x$-axis is labeled as Lemon-lime soda; the $y$-axis is labeled as Cranberry-apple juice.

  • A line is drawn through the plotted points to indicate the general linear trend.

  • Because partial purchases are disallowed, only discrete whole-number grid intersections (corner points) represent actionable choices: $(0, 6)$, $(3, 4)$, $(6, 2)$, and $(9, 0)$.

Real-World Example 2: Volunteer Club Apparel Purchase

• Scenario parameters:

  • T-shirt cost: $10.00\,\text{dollars}$ per shirt.

  • Sweatshirt cost: $20.00\,\text{dollars}$ per sweatshirt.

  • Total budget: $100.00\,\text{dollars}$.

• Variable definitions:

  • Let $x$ = number of t-shirts purchased.

  • Let $y$ = number of sweatshirts purchased.

• Domain and range constraints:

  • Fractional clothing items cannot be purchased; variables must be constrained.

• Equation setup:

  • $10x + 20y = 100$

• Table of evaluated possibilities for $10x + 20y = 100$:

  • x = 0 \formatError{ \to } 10(0) + 20y = 100 \formatError{ \to } 20y = 100 \formatError{ \to } y = 5 (Valid point: $(0, 5)$)

  • x = 1 \formatError{ \to } 10(1) + 20y = 100 \formatError{ \to } 20y = 90 \formatError{ \to } y = 4.5 (Invalid)

  • x = 2 \formatError{ \to } 10(2) + 20y = 100 \formatError{ \to } 20y = 80 \formatError{ \to } y = 4 (Valid point: $(2, 4)$)

  • x = 3 \formatError{ \to } 10(3) + 20y = 100 \formatError{ \to } 20y = 70 \formatError{ \to } y = 3.5 (Invalid)

  • x = 4 \formatError{ \to } 10(4) + 20y = 100 \formatError{ \to } 20y = 60 \formatError{ \to } y = 3 (Valid point: $(4, 3)$)

  • x = 5 \formatError{ \to } 10(5) + 20y = 100 \formatError{ \to } 20y = 50 \formatError{ \to } y = 2.5 (Invalid)

  • x = 6 \formatError{ \to } 10(6) + 20y = 100 \formatError{ \to } 20y = 40 \formatError{ \to } y = 2 (Valid point: $(6, 2)$)

  • x = 7 \formatError{ \to } 10(7) + 20y = 100 \formatError{ \to } 20y = 30 \formatError{ \to } y = 1.5 (Invalid)

  • x = 8 \formatError{ \to } 10(8) + 20y = 100 \formatError{ \to } 20y = 20 \formatError{ \to } y = 1 (Valid point: $(8, 1)$)

  • x = 9 \formatError{ \to } 10(9) + 20y = 100 \formatError{ \to } 20y = 10 \formatError{ \to } y = 0.5 (Invalid)

  • x = 10 \formatError{ \to } 10(10) + 20y = 100 \formatError{ \to } 20y = 0 \formatError{ \to } y = 0 (Valid point: $(10, 0)$)

• Analysis of mathematical restrictions:

  • Substituting odd values for $x$ produces fractional $y$ values ending in $.5$.

  • Domain restriction: $x$ must be restricted to even whole numbers: $\{0, 2, 4, 6, 8, 10\}$.

  • Range restriction: $y$ consists of whole numbers less than or equal to $5$: $\{0, 1, 2, 3, 4, 5\}$.

  • Graphical intersection: Connecting $(0, 5)$ and $(10, 0)$ confirms that grid line intersections occur strictly at even $x$ coordinates.

Academic Tasks and Administrative Requirements

• 3.1 Quick Check Content Summary:

  • Classification: Determining whether equations are linear or non-linear.

  • Real Numbers: Identifying if constants $a$, $b$, and $c$ are real numbers or non-real numbers.

  • Point Testing: Checking if a coordinate point lies on a line by direct algebraic substitution into the equation.

  • Word Problems: Solving real-world contextual problems with provided variable definitions.

• Required Submissions and Course Expectations:

  • Current work: Complete 3.1 Quick Check during class time.

  • Next assignment: 3.1 Practice Plus scheduled for tomorrow.

  • Scratch work: Scratch work must be uploaded for assignments.

  • RTI tracking: Upload recorded RTI minutes completed for the week.

  • Tutoring documentation: Document specific working times and exact topics covered during tutoring sessions.

  • Upcoming Assessment: Module exam on Monday; students must log in using a school-issued device during class time.