Let's solve each of these questions step by step.

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a) Through points (-5, 2) and (4, 1)

The equation of a line can be written in slope-intercept form:
$y = mx + b$ where $m$ is the slope, and $b$ is the y-intercept.

Step 1: Find the slope (m)

The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points $(-5, 2)$ and $(4, 1)$: $m = \frac{1 - 2}{4 - (-5)} = \frac{-1}{4 + 5} = \frac{-1}{9}$

So, the slope $m = -\frac{1}{9}$.

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

Now, we use the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$ Substitute $m = -\frac{1}{9}$ and the point $(-5, 2)$: $y - 2 = -\frac{1}{9}(x + 5)$ Now simplify: $y - 2 = -\frac{1}{9}x - \frac{5}{9}$ $y = -\frac{1}{9}x - \frac{5}{9} + 2$ Convert 2 to a fraction with denominator 9: $y = -\frac{1}{9}x - \frac{5}{9} + \frac{18}{9}$ $y = -\frac{1}{9}x + \frac{13}{9}$

Thus, the equation is: $y = -\frac{1}{9}x + \frac{13}{9}$

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b) Through (-5, 2) parallel to $y = -3x + 6$

Step 1: Identify the slope

Since the line we are looking for is parallel to $y = -3x + 6$, the slope will be the same as the slope of this line. The slope of $y = -3x + 6$ is $m = -3$.

Step 2: Use point-slope form

Using the point-slope form with slope $m = -3$ and the point $(-5, 2)$: $y - 2 = -3(x + 5)$ Simplify: $y - 2 = -3x - 15$ $y = -3x - 15 + 2$ $y = -3x - 13$

Thus, the equation is: $y = -3x - 13$

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c) Through (-5, 2) perpendicular to $y = -x - 1$

Step 1: Identify the slope

The slope of $y = -x - 1$ is $m = -1$. For a line to be perpendicular, the slope must be the negative reciprocal of $-1$, which is $1$.

Step 2: Use point-slope form

Using the point-slope form with slope $m = 1$ and the point $(-5, 2)$: $y - 2 = 1(x + 5)$ Simplify: $y - 2 = x + 5$ $y = x + 5 + 2$ $y = x + 7$

Thus, the equation is: $y = x + 7$

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Let me know if you need more details or clarifications.

Here are 5 related questions:

1. What is the equation of a line passing through (3, 4) and (5, -2)?
2. Find the slope of a line parallel to $y = 2x + 1$.
3. What is the equation of a line perpendicular to $y = 4x - 3$ through point (1, 5)?
4. How do you find the y-intercept of a line given two points?
5. What does it mean geometrically for two lines to be parallel or perpendicular?

Tip: When finding the equation of a line from two points, always start by finding the slope, then use the point-slope form for simplicity.