Determine the point-slope form of a line passing through the points $$ (1, 2) $$ and $$ (3, 8) $$.

Question

UpStudy AI · Accepted Answer

To determine the **point-slope form** of the line passing through the points $$ (1, 2) $$ and $$ (3, 8) $$, follow these steps:

### 1. Find the Slope ($$ m $$) of the Line

The slope of a line between two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is calculated using the formula:

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

Plugging in the given points $$ (1, 2) $$ and $$ (3, 8) $$:

$$
m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3
$$

So, the slope $$ m $$ is **3**.

### 2. Use the Point-Slope Formula

The **point-slope form** of a line is given by:

$$
y - y_1 = m(x - x_1)
$$

You can use either of the two points to write the equation. Let's use the point $$ (1, 2) $$:

$$
y - 2 = 3(x - 1)
$$

### 3. Simplify (Optional)

While the question asks for the **point-slope form**, it's often helpful to simplify or rewrite it in other forms:

**Slope-Intercept Form ($$ y = mx + b $$):**

$$
\begin{align*}
y - 2 &= 3(x - 1) \
y - 2 &= 3x - 3 \
y &= 3x - 3 + 2 \
y &= 3x - 1
\end{align*}
$$

**Standard Form ($$ Ax + By = C $$):**

$$
\begin{align*}
y &= 3x - 1 \
-3x + y &= -1 \
3x - y &= 1 \quad \text{(Multiplying both sides by -1)}
\end{align*}
$$

### 4. Final Answer

The **point-slope form** of the line passing through $$ (1, 2) $$ and $$ (3, 8) $$ is:

$$
y - 2 = 3(x - 1)
$$

This equation accurately represents the line through the given points.
The point-slope form of the line is $$ y - 2 = 3(x - 1) $$.

Determine the point-slope form of a line passing through the points \( (1, 2) \) and \( (3, 8) \).