A line passes through $(x_1, y_1)$ and $(h, k)$. If slope of the line is $m$, show that $k - y_1 = m(h - x_1)$.

Question

Accepted Answer

> [!TLDR]
> The slope formula $m = \frac{\Delta y}{\Delta x}$ directly gives $k - y_1 = m(h - x_1)$ when rearranged — this is the point-slope form of a line.

## Why This Works

The slope of a line is a measure of its steepness — how much the $y$-coordinate changes for each unit change in $x$. If you know two points on the line, the slope is simply the ratio of the vertical change to the horizontal change. That's the core idea here.

When we say a line passes through $(x_1, y_1)$ and $(h, k)$ with slope $m$, we are essentially saying: *the ratio of the difference in $y$ to the difference in $x$ between these two points equals $m$*. The equation $k - y_1 = m(h - x_1)$ is just that statement, rearranged.

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## Step-by-Step Derivation

1. **Recall the definition of slope.**  
   For any two points $(x_1, y_1)$ and $(x_2, y_2)$ on a non-vertical line, the slope $m$ is given by:
   $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
   This is the *rise* (change in $y$) divided by the *run* (change in $x$).

2. **Identify the two points in the problem.**  
   Here, the two points are $(x_1, y_1)$ and $(h, k)$. So we set:
   $$x_2 = h, \quad y_2 = k$$

3. **Substitute into the slope formula.**  
   Plugging these into the definition:
   $$m = \frac{k - y_1}{h - x_1}$$

> [!WARNING]
   > A common mistake is to swap the order — e.g., writing $\frac{y_1 - k}{x_1 - h}$. That gives the same numerical value (since both numerator and denominator flip sign), but it's safer to keep the order consistent: second point minus first point.

4. **Multiply both sides by $(h - x_1)$.**  
   To isolate the relationship between $k$, $y_1$, $h$, and $x_1$, multiply through:
   $$m(h - x_1) = k - y_1$$

5. **Rewrite in the required form.** …

Slope value	What the line does
$m > 0$	Rises left to right (uphill)
$m < 0$	Falls left to right (downhill)
$m = 0$	Horizontal (flat)
$m$ undefined	Vertical (straight up/down)

Q.A line passes through $(x_1, y_1)$ and $(h, k)$ . If slope of the line is $m$ , show that $k - y_1 = m(h - x_1)$ .

Concept understanding — Slope Calculation

Slope Calculation — From Intuition to Precision

The Intuition: Rise Over Run

The Precise Definition

What the Number Tells You

A Worked Example

Why This Works

Step-by-Step Derivation

Q.A line passes through (x1,y1)(x_1, y_1)(x1​,y1​) and (h,k)(h, k)(h,k). If slope of the line is mmm, show that k−y1=m(h−x1)k - y_1 = m(h - x_1)k−y1​=m(h−x1​).

Why This Works

Step-by-Step Derivation

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Q.A line passes through $(x_1, y_1)$ and $(h, k)$ . If slope of the line is $m$ , show that $k - y_1 = m(h - x_1)$ .

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