SAT-Math-6: Calculating Slope ($m = \frac{y_2 - y_1}{x_2 - x_1}$) from Two Points

1. The Slope Formula

When you are given any two points on a coordinate plane, $(x_1, y_1)$ and $(x_2, y_2)$, you can find the exact slope ($m$) by calculating the vertical change divided by the horizontal change.

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

This is often called the "change in $y$ over the change in $x$":

$$m = \frac{\Delta y}{\Delta x}$$

🇺🇿 Uzbek Explanation:

Ikki nuqtadan o'tuvchi chiziq nishabligi: Agar sizga koordinata tekisligida ikkita nuqta berilgan bo'lsa, nishablikni ($m$) topish uchun $y$ koordinatalar ayirmasini $x$ koordinatalar ayirmasiga bo'lishingiz kerak. Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

2. The Danger Zone: The Negative Sign Trap

The absolute most common mistake students make with this formula is losing track of negative signs. If a coordinate itself is negative, subtracting it creates a double negative, which becomes addition!

$$\text{Subtracting a negative: } y_2 - (-y_1) = y_2 + y_1$$

3. Step-by-Step High-Yield Examples

Example 1: Standard Calculation with Negatives

Question: Find the slope of the line that passes through the points $(-2, 4)$ and $(3, -6)$.

Step-by-Step Breakdown:

1. Label your points clearly so you don't mix them up:

   • $(x_1, y_1) = (-2, 4)$

   • $(x_2, y_2) = (3, -6)$

2. Plug them into the formula:

   $$m = \frac{-6 - 4}{3 - (-2)}$$

3. Simplify carefully:

   • Top (Numerator): $-6 - 4 = -10$

   • Bottom (Denominator): $3 - (-2) = 3 + 2 = 5$

4. Final division:

   $$m = \frac{-10}{5} = -2$$

Example 2: Finding a Missing Coordinate (SAT Style)

Question: A line passing through the points $(1, 4)$ and $(5, k)$ has a slope of $\frac{3}{2}$. What is the value of $k$?

Step-by-Step Breakdown:

1. Set up the formula with the given information:

   $$m = \frac{y_2 - y_1}{x_2 - x_1} \implies \frac{3}{2} = \frac{k - 4}{5 - 1}$$

2. Simplify the denominator:

   $$\frac{3}{2} = \frac{k - 4}{4}$$

3. Cross-multiply to solve the proportion:

   $$3 \cdot 4 = 2 \cdot (k - 4)$$

   $$12 = 2k - 8$$

4. Isolate $k$:

   Add $8$ to both sides: $20 = 2k$

   Divide by $2$: $10 = k$

Final Answer: $k = 10$

Summary

• Order matters: If you start with $y_2$ on top, you must start with $x_2$ on the bottom.

• Watch for double negatives: Subtracting a negative coordinate turns into addition ($-\ (-3) = +3$).

• Fractions are fine: SAT slopes are frequently left as simplified improper fractions (like $\frac{3}{2}$) rather than decimals.