To find the $x$-intercept of the line, we need to identify the point where the line crosses the $x$-axis. This occurs when $y = 0$.

Looking at the given points, we have:

• $(33, -22)$
• $(52, -33)$
• $(71, -44)$

None of these points have $y = 0$, so the line doesn't cross the $x$-axis at these points.

We can use the equation of the line (in slope-intercept form, $y = mx + b$) to determine the $x$-intercept.

Steps to find the $x$-intercept:

1. Find the slope of the line (m): We can use two points, say $(33, -22)$ and $(52, -33)$, to calculate the slope $m$.

   $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-33 - (-22)}{52 - 33} = \frac{-33 + 22}{52 - 33} = \frac{-11}{19}$

   So, the slope is $m = \frac{-11}{19}$.

2. Use the point-slope form to find the equation of the line: The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Using point $(33, -22)$ and the slope $m = \frac{-11}{19}$, we substitute into the equation: $y - (-22) = \frac{-11}{19}(x - 33)$ Simplify: $y + 22 = \frac{-11}{19}(x - 33)$ Distribute the slope on the right side: $y + 22 = \frac{-11}{19}x + \frac{11 \times 33}{19}$ Simplifying: $y + 22 = \frac{-11}{19}x + \frac{363}{19}$ Now, solve for $y$: $y = \frac{-11}{19}x + \frac{363}{19} - 22$ Convert 22 to a fraction with denominator 19: $22 = \frac{418}{19}$ Substituting: $y = \frac{-11}{19}x + \frac{363}{19} - \frac{418}{19}$ $y = \frac{-11}{19}x - \frac{55}{19}$

3. Find the $x$-intercept: The $x$-intercept occurs when $y = 0$. Set $y = 0$ in the equation: $0 = \frac{-11}{19}x - \frac{55}{19}$ Multiply both sides by 19 to eliminate the denominator: $0 = -11x - 55$ Solve for $x$: $11x = -55$ $x = -5$

Thus, the $x$-intercept of the line is at $x = -5$.

Would you like a more detailed breakdown of any part of this process? Or do you have any follow-up questions?

Related Questions:

1. How would the equation change if a different pair of points were used?
2. How do you find the slope when both coordinates of one point are negative?
3. How does the slope affect the steepness of a line?
4. What happens if the slope is 0? How does that change the equation?
5. How can you verify the line equation using a third point on the line?

Tip: To quickly check your slope calculation, ensure that the change in $y$ is divided by the change in $x$, not the other way around!