To find the area of the triangle formed by the line $\ell$ and the coordinate axes, we start by analyzing the given slope and determining the points where the line intersects the axes.

Step 1: Interpret the Slope and Formulate the Line Equation

The problem states that the slope of line $\ell$ is $3$. Therefore, we know: $\text{slope} = 3$ The general equation for a line with slope $m$ is: $y = mx + b$ Since the slope $m = 3$, we have: $y = 3x + b$ To find the $y$-intercept $b$, we use the fact that the line passes through the point $(2, 0)$ as shown in the figure.

Step 2: Substitute the Point $(2, 0)$ to Find $b$

Substitute $x = 2$ and $y = 0$ into the line equation: $0 = 3(2) + b$ $0 = 6 + b$ $b = -6$ Thus, the equation of line $\ell$ is: $y = 3x - 6$

Step 3: Find the Intercepts

To find the points where the line intersects the $x$- and $y$-axes, set $y = 0$ for the $x$-intercept and $x = 0$ for the $y$-intercept.

$x$-Intercept

Set $y = 0$: $0 = 3x - 6$ $3x = 6$ $x = 2$ So, the $x$-intercept is $(2, 0)$.

$y$-Intercept

Set $x = 0$: $y = 3(0) - 6$ $y = -6$ So, the $y$-intercept is $(0, -6)$.

Step 4: Calculate the Area of the Triangle

The triangle formed by the line and the coordinate axes has vertices at $(0, 0)$, $(2, 0)$, and $(0, -6)$.

The base of the triangle is the distance along the $x$-axis from $(0, 0)$ to $(2, 0)$, which is $2$ units.

The height of the triangle is the distance along the $y$-axis from $(0, 0)$ to $(0, -6)$, which is $6$ units.

The area $A$ of a triangle is given by: $A = \frac{1}{2} \times \text{base} \times \text{height}$ Substitute the values for the base and height: $A = \frac{1}{2} \times 2 \times 6 = \frac{1}{2} \times 12 = 6$

Conclusion

The area of the triangle formed by the line and the coordinate axes is: $\boxed{6}$

Would you like a breakdown of any steps or additional details?

Here are some related questions you might find interesting:

1. How do you find the equation of a line given two points?
2. What is the formula for the area of a triangle with vertices at specific points?
3. How does changing the slope affect the intercepts of a line?
4. How can you determine the area of a shape bounded by a line and the axes in the coordinate plane?
5. What happens to the area of the triangle if the slope changes but the intercept remains fixed?

Tip: Knowing how to find intercepts quickly helps in determining areas of shapes in the coordinate plane!