Let's solve the problem step by step.

Part (a): Write the equation of the line

We are given:

• A point on the line: $(2, 5)$
• The slope of the line: $m = 3$

To find the equation of a line given a point $(x_1, y_1)$ and a slope $m$, we use the point-slope form of the line equation:

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

Substituting the given point $(x_1, y_1) = (2, 5)$ and slope $m = 3$:

$y - 5 = 3(x - 2)$

Now, distribute the slope and simplify the equation:

$y - 5 = 3x - 6$

Add 5 to both sides to solve for $y$:

$y = 3x - 1$

Thus, the equation of the line is:

$y = 3x - 1$

Part (b): Sketch the graph

To sketch the graph of the line $y = 3x - 1$, follow these steps:

1. Identify the slope and y-intercept:

   • Slope ($m$) is 3, which means the line rises 3 units for every 1 unit it moves to the right.
   • Y-intercept ($b$) is -1, so the line crosses the y-axis at $(0, -1)$.

2. Plot the y-intercept:

   • Start by plotting the point $(0, -1)$ on the graph.

3. Use the slope to plot another point:

   • From $(0, -1)$, move up 3 units (since the slope is 3) and 1 unit to the right. This gives you another point on the line: $(1, 2)$.

4. Draw the line:

   • Draw a straight line through the points $(0, -1)$ and $(1, 2)$. This line represents the equation $y = 3x - 1$.

Would you like more details, or do you have any questions?

Additional Questions for Practice:

1. What is the equation of a line passing through the point $(4, -3)$ with a slope of $-2$?
2. How do you convert a point-slope form equation to a slope-intercept form equation?
3. If a line has a slope of 0, what does its graph look like?
4. Can two different lines have the same slope but different y-intercepts? Explain why or why not.
5. How would you find the x-intercept of the line $y = 3x - 1$?

Tip:

To find the equation of a line quickly, remember that the slope-intercept form is $y = mx + b$, where $m$ is the slope, and $b$ is the y-intercept.