Find the equation of the line. Write your answer as exact expression with fractions if necessary.

a) through (-5,2) and (4,1)

y=   

b) through (-5,2) parallel to y=-3x+6

y=   

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

y=   

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:
\[
y = -\frac{1}{9}x + \frac{13}{9}
\]

---

### 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:
\[
y = -3x - 13
\]

---

### 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:
\[
y = x + 7
\]

---

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.

Find the equation of the line. Write your answer as exact expression with fractions if necessary.

a) through (-5,2) and (4,1)

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

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

Learn how to find the equation of a line through given points, parallel, or perpendicular to another line. This step-by-step guide covers slope-intercept form, point-slope form, and the properties of parallel and perpendicular lines. Suitable for Grades 8-10.

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