To find the $x$-intercept of the line from the given table of points, we need to follow these steps:

1. Determine the slope ($m$) of the line: The formula for slope is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Let's use two points, $(-38, 40)$ and $(-23, 30)$: $m = \frac{30 - 40}{-23 - (-38)} = \frac{-10}{15} = -\frac{2}{3}.$

2. Find the equation of the line: The slope-intercept form of a line is: $y = mx + b$ Substitute one point, say $(-38, 40)$, and $m = -\frac{2}{3}$ to solve for $b$ (the $y$-intercept): $40 = -\frac{2}{3}(-38) + b.$ Simplify: $40 = \frac{76}{3} + b.$ Subtract $\frac{76}{3}$ from 40: $b = 40 - \frac{76}{3} = \frac{120}{3} - \frac{76}{3} = \frac{44}{3}.$ So the equation of the line is: $y = -\frac{2}{3}x + \frac{44}{3}.$

3. Find the $x$-intercept: At the $x$-intercept, $y = 0$. Substitute $y = 0$ into the equation: $0 = -\frac{2}{3}x + \frac{44}{3}.$ Solve for $x$: $\frac{2}{3}x = \frac{44}{3}.$ Multiply through by 3: $2x = 44.$ Divide by 2: $x = 22.$

Final Answer:

The $x$-intercept of the line is $x = 22$.

Would you like a detailed explanation of any step?

Related Questions:

1. How can we derive the slope-intercept equation from two points on a line?
2. What is the significance of the $x$-intercept in real-world problems?
3. How do we calculate the $y$-intercept directly from two points?
4. What changes in the process if the line is vertical or horizontal?
5. How does the slope of a line affect its graphical representation?

Tip:

Always double-check your slope calculations when finding intercepts—small errors can lead to incorrect results!