This worked example demonstrates how to derive a linear equation by calculating slope and solving for the y-intercept using coordinate points.

You've provided a clear and correct step-by-step solution to find the equation of a line passing through two points: (3, 4) and (5, 8). Let's go through it carefully and verify each step, then explain why it works.

---

✔ Problem:

Find the equation of the line that passes through the points (3, 4) and (5, 8).

---

🔹 Step 1: Calculate the Slope

The formula for slope $ m $ between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is:

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

Using the points (3, 4) and (5, 8):

$$
m = \frac{8 - 4}{5 - 3} = \frac{4}{2} = 2
$$

✔ Correct. The slope is 2.

---

🔹 Step 2: Use the Slope-Intercept Form

The slope-intercept form of a line is:

$$
y = mx + b
$$

We now know $ m = 2 $, so:

$$
y = 2x + b
$$

✔ Correct.

---

🔹 Step 3: Plug in One Point to Solve for $ b $

Use one of the given points — let's use (3, 4). Plug $ x = 3 $, $ y = 4 $ into the equation:

$$
4 = 2(3) + b
$$

$$
4 = 6 + b
$$

✔ Correct.

---

🔹 Step 4: Solve for $ b $

$$
4 = 6 + b
\Rightarrow b = 4 - 6 = -2
$$

✔ Correct. So, $ b = -2 $

---

🔹 Step 5: Write the Final Equation

Now substitute $ m = 2 $ and $ b = -2 $ into $ y = mx + b $:

$$
y = 2x - 2
$$

✔ Correct.

---

✔ Final Answer:

$$
\boxed{y = 2x - 2}
$$

---

📌 Explanation Summary:

- Slope tells us how steep the line is. A slope of 2 means for every 1 unit increase in $ x $, $ y $ increases by 2.
- We used the slope-intercept form because we wanted the equation in terms of $ y = mx + b $, which is standard.
- By plugging in a known point, we solved for the y-intercept ($ b $) — the value of $ y $ when $ x = 0 $.
- This gives us a complete equation that describes all points on the line.

---

🔍 Optional Check: Verify with the Second Point

Let’s test if (5, 8) satisfies the equation $ y = 2x - 2 $:

$$
y = 2(5) - 2 = 10 - 2 = 8
$$

✔ Yes! It matches. So the line passes through both points.

---

✔ Conclusion:

Your solution is entirely correct and well-structured. Great job!

Final Answer:
$$
\boxed{y = 2x - 2}
$$