We will first find the equation of the line passing through the points $(2, -2)$ and $(-2, 6)$.

Step 1: Find the slope $m$

The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates $(2, -2)$ and $(-2, 6)$:

$m = \frac{6 - (-2)}{-2 - 2} = \frac{6 + 2}{-4} = \frac{8}{-4} = -2$

Step 2: Use the point-slope form to find the equation of the line

The point-slope form of a line is:

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

Substituting $m = -2$ and one of the points, say $(2, -2)$:

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

Simplifying:

$y + 2 = -2(x - 2)$ $y + 2 = -2x + 4$ $y = -2x + 2$

Thus, the equation of the line is:

$y = -2x + 2$

Step 3: Find the x-intercept and y-intercept

1. Y-intercept: The y-intercept is where the line crosses the y-axis, which occurs when $x = 0$. Substituting $x = 0$ into the equation:

$y = -2(0) + 2 = 2$

Thus, the y-intercept is $(0, 2)$.

2. X-intercept: The x-intercept is where the line crosses the x-axis, which occurs when $y = 0$. Setting $y = 0$ in the equation:

$0 = -2x + 2$

Solving for $x$:

$2x = 2 \quad \Rightarrow \quad x = 1$

Thus, the x-intercept is $(1, 0)$.

Final Answer:

• The equation of the line is: $y = -2x + 2$
• The x-intercept is: $(1, 0)$
• The y-intercept is: $(0, 2)$

Would you like more details or have any questions?

Here are some related questions for you to explore:

1. How do you find the slope of a line if one point is at infinity?
2. What is the general form of a linear equation, and how does it relate to slope-intercept form?
3. How do you calculate the distance between two points on a line?
4. What is the significance of the intercepts in graphing linear equations?
5. Can a line have more than one x-intercept or y-intercept?

Tip: Always double-check your intercepts by substituting the values back into the original equation to verify they satisfy the equation!