To find the equation of the line with a gradient of $3$ that passes through the point $(2, 10)$, we can use the slope-intercept form of a line:

$y = mx + c$

where:

• $m$ is the gradient (slope),
• $c$ is the y-intercept.

Step-by-Step Solution

1. Identify the known values:

   • Gradient ($m$) is $3$.
   • The line passes through the point $(2, 10)$.

2. Substitute into the equation: Substitute $m = 3$, $x = 2$, and $y = 10$ into the equation to solve for $c$:

   $10 = 3 \cdot 2 + c$

3. Calculate $c$: $10 = 6 + c$ $c = 10 - 6 = 4$

4. Write the equation: Now that we know $m = 3$ and $c = 4$, the equation of the line is:

   $y = 3x + 4$

Final Answer

The equation of the line is:

$y = 3x + 4$

Would you like a more detailed breakdown or have any questions?

Related Questions

1. How would the equation change if the line passed through a different point with the same gradient?
2. What is the equation of a line with a gradient of -3 that passes through the point (2, 10)?
3. How can we find the equation of a line if we know two points on the line instead of the gradient?
4. How would the equation look if the line was parallel to $y = 3x + 4$ but had a different y-intercept?
5. What are some real-life examples where calculating the equation of a line is useful?

Tip

Remember, you can always find the y-intercept by plugging in the coordinates of a known point on the line into the equation $y = mx + c$.