Algebraic Methods for Finding Linear Equations from Two Points

The primary objective is to determine the equation of a line when provided with two specific points (ordered pairs), without relying on a pre-drawn graph or grid paper.
Historically, finding the slope involved visual tools or iterative processes:
- The Slope Triangle Method: On a physical graph, the slope is determined by creating a right-angled triangle between two points to measure the "change in $y$ " (rise) over the "change in $x$ " (run).
- The Table Method: Points are placed in an $x/y$ table to observe the constant rate of change. For example, if moving from point $(3, 4)$ to $(5, 8)$ , the change in $y$ is $+4$ and the change in $x$ is $+2$ , leading to a slope of: $m = \frac{4}{2} = 2$
The Problem with Working Backwards: To find the "starting point" (the $(y)$ -intercept, or $b$ ), one could previously work backwards on a table (e.g., following the pattern from $x=3$ to $x=2, 1, 0$ ). However, this is inefficient and time-consuming if the points are far apart or if no grid is available.
The Algebraic Transition: The modern, more practical approach is to solve the problem algebraically using numbers, equations, and variables rather than completing extensive tables or manual graphing.

The goal is to produce an equation in the form of $y = mx + b$ .
Step 1: Determine the Slope ( $m$ ):
- Line up the two points vertically as if they were in a table to find the differences.
- Calculate the unit rate of change: $m = \frac{\text{change in } y}{\text{change in } x}$ .
- It does not matter which point is placed first or second, provided the direction of subtraction is consistent for both $x$ and $y$ .
Step 2: Solve for the $y$ -intercept ( $b$ ):
- Recognize that an equation cannot be solved if it has too many unknown variables (e.g., an equation with $x$ , $y$ , and $b$ cannot be solved for $b$ simultaneously).
- To isolate $b$ , substitute the calculated value for $m$ and the coordinates from one of the given points $(x, y)$ into the standard equation $y = mx + b$ .
- Once the values for $y$ , $m$ , and $x$ are plugged in, only one variable ( $b$ ) remains, making the equation solvable.
Step 3: Finalize the Equation:
- After finding the value of $b$ , write the final equation, keeping $y$ and $x$ as variables.

Finding the Slope:
- Change in $y$ : From $4$ to $8$ is $+4$ .
- Change in $x$ : From $3$ to $5$ is $+2$ .
- $m = \frac{4}{2} = 2$
Substituting to Find $b$ (Option 1: Point $(3, 4)$ ):
- Set up the equation: $4 = 2 \times 3 + b$
- Combine like terms: $4 = 6 + b$
- Isolate $b$ : $4 - 6 = b \Rightarrow b = -2$
Substituting to Find $b$ (Option 2: Point $(5, 8)$ ):
- Set up the equation: $8 = 2 \times 5 + b$
- Combine like terms: $8 = 10 + b$
- Isolate $b$ : $8 - 10 = b \Rightarrow b = -2$
Conclusion: The choice of point does not affect the outcome. The final equation is: $y = 2x - 2$

Finding the Slope:
- Change in $y$ : From $5$ to $3$ is $-2$ .
- Change in $x$ : From $4$ to $8$ is $+4$ .
- $m = \frac{-2}{4} = -0.5$ (or $-\frac{1}{2}$ ).
Order Consistency: If the points were reversed ( $(8, 3)$ then $(4, 5)$ ):
- Change in $y$ : From $3$ to $5$ is $+2$ .
- Change in $x$ : From $8$ to $4$ is $-4$ .
- $m = \frac{2}{-4} = -0.5$ . The result remains identical.
Substituting to Find $b$ (Using point $(4, 5)$ ):
- Equation: $5 = -0.5 \times 4 + b$
- Combine like terms ( $-0.5 \times 4$ is half of $4$ , which is $2$ , then make it negative): $5 = -2 + b$
- Isolate $b$ : Add $2$ to both sides: $5 + 2 = b \Rightarrow b = 7$
Final Equation: $y = -0.5x + 7$

Finding the Slope:
- Change in $y$ : From $2$ to $10$ is $+8$ .
- Change in $x$ : From $1$ to $5$ is $+4$ .
- $m = \frac{8}{4} = 2$
Substituting to Find $b$ (Using point $(1, 2)$ ):
- Equation: $2 = 2 \times 1 + b$
- Combine like terms: $2 = 2 + b$
- Isolate $b$ : $2 - 2 = b \Rightarrow b = 0$
Implications of $b = 0$ :
- A $y$ -intercept of zero means there is no numerical constant added to the end of the equation.
- The line passes directly through the origin $(0, 0)$ .
- The starting point is zero.
Final Equation: $y = 2x$

Matayi's Inquiry: Matayi asked for clarification on how to solve for $b$ . The instructor explained that you must write an equation where only $b$ is missing by using the existing $x$ and $y$ coordinates from the points provided.
Student Point Preference: A student asked if it matters which point is chosen for substitution. The instructor demonstrated through Example 1 (points $(3, 4)$ and $(5, 8)$ ) that both points yield the same $b$ value ( $-2$ ).
Bryce's Interaction: Bryce was called upon to identify the calculated slope during Example 2. The instructor emphasized the importance of this algebraic process for next year's curriculum and noted that identifying $m$ is the first half of the work.
Deshaun's Participation: Deshaun collaborated on checking the equation steps, treated as a "coworker" in the problem-solving process.
Miscellaneous: During the independent practice for the final example, there was brief tangential dialogue regarding lunch options, specifically "loaded fries" versus "pizza," with a suggestion to ask for a bite of the fries to determine if they are "mid" (average).
Homework Request: Students requested a short homework assignment; the instructor agreed to put it together so they could start working immediately.