Equations in Two Variables and Linear Functions

Linear Functions and Equations Fundamentals

• Overview of linear functions, equations in two variables, slope, intercepts, forms of linear lines, horizontal and vertical lines, and parallel/perpendicular lines.

Definition and Calculation of Slope

• Slope ($m$) represents the steepness and direction of a line, defined as the ratio of vertical change (rise) to horizontal change (run).

• Mathematical representations of slope:   $\text{slope} = m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$

• The notation $\frac{\Delta y}{\Delta x}$ is read as "delta $y$ over delta $x$" or "change in $y$ over change in $x$".

• Format and expression rules for slope values:

  • Slope must always be expressed as a fraction in simplest form. Improper fractions are acceptable (e.g., $\frac{4}{3}$).

  • Exception: In real-life context problems, a decimal value for slope is acceptable.

  • In real-life scenarios, slope is often referred to as the rate of change and can be identified by the word "per".

Real-World Application: Rate of Change in Animal Shelter Populations

• Contextual scenario: In $2015$, there were an estimated $7.6\,\text{million}$ animals (dogs and cats) in shelters across the United States. By $2020$, this number had dropped to $4.3\,\text{million}$.

• Variable definitions:

  • Let $x$ represent the number of years since $2015$ (where $x = 0$ corresponds to $2015$ and $x = 5$ corresponds to $2020$).

  • Let $y$ represent the shelter population in millions of animals.

• Step-by-step rate of change calculation:   $\text{Rate of change} = \frac{y_2 - y_1}{x_2 - x_1}$   $\text{Rate of change} = \frac{4.3 - 7.6}{5 - 0}$   $\text{Rate of change} = \frac{-3.3}{5}$   $\text{Rate of change} \approx -0.66\,\text{million animals per year}$

• Real-world interpretation:

  • The shelter population decreased by approximately $0.66\,\text{million}$ animals per year, which corresponds to a reduction of approximately $660,000\,\text{animals per year}$ from $2015$ to $2020$.

Distinction Between Linear Functions and Linear Equations

• Linear Function:

  • Function notation: $f(x) = mx + b$

  • Graphical representation: Graphs as ordered pairs in the form $(x, f(x))$

• Linear Equation:

  • Equation notation: $y = mx + b$

  • Solution representation: Has solutions in the form of ordered pairs $(x, y)$

Slope-Intercept Form

• Slope-intercept form (also known as function form when expressed using $f(x)$):   $y = mx + b$   $f(x) = mx + b$

• Component analysis:

  • $m$ represents the slope of the line (the coefficient of $x$).

  • $b$ represents the $y$-intercept.

  • $x$ and $y$ stay "open" as variables.

Real-World Application: Hiking Rate and Predictions

• Problem scenario: A graph shows the distance a friend hiked in $x$ hours, passing through $(0, 0)$ and $(7, 24.5)$.

• Part 1: Calculating the hiking rate and writing the equation:

  • Slope calculation:     $m = \frac{24.5 - 0}{7 - 0}$     $m = \frac{24.5}{7}$     $m = 3.5$

  • Interpretation: The friend hikes at a rate of $3.5\,\text{miles per hour}$.

  • Equation representation:     $d = 3.5t$     where $d$ represents distance in miles and $t$ represents time in hours.

• Part 2: Determining checkpoint arrival time:

  • Problem statement: The friend started hiking at $8:30\,\text{a.m.}$ with a scheduled checkpoint $11\,\text{miles}$ from the starting point.

  • Calculation:     $d = 3.5t$     $11 = 3.5t$     $t = \frac{11}{3.5}$     $t \approx 3.14\,\text{hours}$

  • Time conversion:

  • $3.14\,\text{hours}$ corresponds to $3\,\text{hours}$ and $0.14 \times 60 \approx 8.4\,\text{minutes}$.

  • Adding $3\,\text{hours}$ and $8.4\,\text{minutes}$ to $8:30\,\text{a.m.}$ gives approximately $11:38\,\text{a.m.}$

  • Conclusion: The friend should reach the checkpoint approximately $3.14\,\text{hours}$ after starting the hike, at approximately $11:38\,\text{a.m.}$

Standard Form of a Linear Equation

• Definition:   $Ax + By = C$

• Conditions for Standard Form:

  • $A$, $B$, and $C$ are integer values (non-fractions).

  • $A$, $B$, and $C$ have no common factor other than $1$

  • $A > 1$

Intercepts of Linear Equations

• $x$-intercept:

  • Definition: The point where a line passes through the horizontal $x$-axis.

  • Value condition: The value of $x$ for which $y = 0$

  • Coordinate notation: Written as the ordered pair $(x, 0)$

• $y$-intercept:

  • Definition: The point where a line passes through the vertical $y$-axis.

  • Value condition: The value of $y$ for which $x = 0$

  • Coordinate notation: Written as the ordered pair $(0, y)$ or $(0, b)$

Procedure for Graphing a Linear Equation Using Intercepts

• Step-by-step procedure:

  1. Write the linear equation in standard form ($Ax + By = C$) if necessary.

  2. Plug zero in for $x$ ($x = 0$), solve for $y$, and write as the ordered pair $(0, y)$.

  3. Plug zero in for $y$ ($y = 0$), solve for $x$, and write as the ordered pair $(x, 0)$.

  4. Graph these two points on the coordinate plane and connect them with a continuous line with arrows at both ends.

• Worked Example: Sketching the graph of $3x - 2y = 4$:

  • Finding the $y$-intercept ($x = 0$):     $3(0) - 2y = 4$     $-2y = 4$     $y = -2$     $y\text{-intercept} = (0, -2)$

• Finding the $x$-intercept ($y = 0$):     $3x - 2(0) = 4$     $3x = 4$     $x = \frac{4}{3}$     $x\text{-intercept} = \left(\frac{4}{3}, 0\right)$

Point-Slope Form and Writing Equations of Lines

• Selection rules when writing an equation given two points:

  • Use slope-intercept (function) form ($y = mx + b$) if given the coordinates of the $y$-intercept (i.e., given $b$ or if the line passes through $(0, b)$).

  • Use point-slope form if neither of the given points is the $y$-intercept.

• Point-Slope Form Definition:   $y - y_1 = m(x - x_1)$   where:

  • $m$ represents the slope of the line:     $m = \frac{y_2 - y_1}{x_2 - x_1}$

  • $(x_1, y_1)$ represents any known point on the line.

  • $x$ and $y$ stay "open" as variables.

• Worked Example: Line passing through $(-2, -6)$ and $(4, 6)$:

  • Step 1: Calculate slope $m$:     $m = \frac{6 - (-6)}{4 - (-2)}$     $m = \frac{6 + 6}{4 + 2}$     $m = \frac{12}{6}$     $m = 2$

  • Step 2: Write in Point-Slope Form:

  • Using point $(-2, -6)$:       $y - (-6) = 2(x - (-2))$       $y + 6 = 2(x + 2)$

  • Using point $(4, 6)$:       $y - 6 = 2(x - 4)$

  • Step 3: Convert to Slope-Intercept Form:

  • From $y + 6 = 2(x + 2)$:       $y + 6 = 2x + 4$       $y = 2x - 2$

  • From $y - 6 = 2(x - 4)$:       $y - 6 = 2x - 8$       $y = 2x - 2$

Horizontal and Vertical Lines

• Horizontal Lines:

  • IS a function.

  • Has a slope of zero ($m = 0$).

  • General equation form:     $y = 0x + b$     $y = b$     where $b$ is the constant $y$-coordinate of every point lying on the horizontal line.

  • Worked Example: Equation of a horizontal line passing through $(7, -10)$:     $y = -10$

• Vertical Lines:

  • Is NOT a function.

  • Cannot take the form $y = mx + b$

  • Slope is undefined ($m = \text{undefined}$, or division by zero $\frac{\text{number}}{0}$).

  • General equation form:     $x = \text{number}$     where the number is the constant $x$-coordinate of every point lying on the vertical line.

  • Worked Example: Equation of a vertical line passing through $(7, -10)$:     $x = 7$

Parallel and Perpendicular Lines

• Parallel Lines:

  • Have the same slope ($m_1 = m_2$).

  • Have different $y$-intercepts ($b_1 \neq b_2$).

• Perpendicular Lines:

  • Have slopes that are negative reciprocals of each other:     $m_1 \rightarrow -\frac{1}{m_1}$     $m_1 \cdot m_2 = -1$

  • Can have the same or different $y$-intercepts.

• Worked Example: Writing an equation for line $l$:

  • Problem statement: Line $l$ passes through $(-3, 2)$ and its graph is perpendicular to the graph of $y = \frac{2}{3}x + 8$. Write a function to describe line $l$.

  • Step 1: Determine the slope of line $l$:

  • Given line slope $m_1 = \frac{2}{3}$.

  • Line $l$ slope $m_2 = -\frac{3}{2}$.

  • Step 2: Use point-slope form with $(-3, 2)$ and $m = -\frac{3}{2}$:     $y - 2 = -\frac{3}{2}(x - (-3))$     $y - 2 = -\frac{3}{2}(x + 3)$     $y - 2 = -\frac{3}{2}x - \frac{9}{2}$

  • Step 3: Rearrange into slope-intercept form:     $y = -\frac{3}{2}x - \frac{9}{2} + 2$     $y = -\frac{3}{2}x - \frac{5}{2}$

  • Step 4: Write in function form:     $f(x) = -\frac{3}{2}x - \frac{5}{2}$