This algebra worksheet helps students practice graphing linear equations on coordinate planes while exploring relationships between parallel lines and slopes.

Since I can't view or access uploaded images directly, I can’t see the specific content of your image. However, based on the text you've provided, it looks like a standard math worksheet involving graphing lines, analyzing their properties, and writing equations of parallel lines.

Let me solve each problem step by step as if I could see the image. If there's anything missing or different in your image, please let me know!

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1. Graph both lines on the same coordinate plane:

a) $ y = 2x - 3 $
b) $ y = -3x + 1 $

#### Graphing:
- For $ y = 2x - 3 $:
Slope = 2 (rise 2, run 1), y-intercept = –3 → starts at (0, –3)
- For $ y = -3x + 1 $:
Slope = –3 (rise –3, run 1), y-intercept = 1 → starts at (0, 1)

#### What do you notice about the two lines?
They intersect at one point because they have different slopes. They are not parallel and are not the same line.

#### What do you notice about the equations for the lines?
The slopes are different (2 vs. –3), so the lines are not parallel. The y-intercepts are also different.

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2. Graph both lines on the same coordinate plane:

a) $ 4y = x - 8 $
b) $ y = \frac{1}{4}x - 2 $

First, rewrite equation (a) in slope-intercept form:
$$
4y = x - 8 \Rightarrow y = \frac{1}{4}x - 2
$$

So both equations become:
- $ y = \frac{1}{4}x - 2 $
- $ y = \frac{1}{4}x - 2 $

They are identical equations.

#### What do you notice about the two lines?
They are the same line — they overlap completely.

#### What do you notice about the equations for the lines?
After simplifying, both equations are identical. So they represent the same line.

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3. Draw a conclusion that summarizes what you observed in the previous exercises.

> Conclusion:
> Two lines are the same if they have the same slope and same y-intercept.
> If two lines have different slopes, they intersect at exactly one point.
> If two lines have the same slope but different y-intercepts, they are parallel and never intersect.
> If two lines have same slope and same y-intercept, they are the same line (coincident).

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4. What is the slope of a line that is parallel to the line $ y = -2x $?

A line parallel to $ y = -2x $ must have the same slope.

Slope of given line: $ m = -2 $

✔ Answer: $ \boxed{-2} $

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5. Write the equation for a line that is parallel to the line $ y = 5 - \frac{1}{2}x $ and passes through the point $ (-6, 4) $.

Step 1: Identify the slope of the given line.

Rewriting: $ y = -\frac{1}{2}x + 5 $ → slope $ m = -\frac{1}{2} $

Parallel line has same slope: $ m = -\frac{1}{2} $

Step 2: Use point-slope form:

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

Plug in $ m = -\frac{1}{2} $, $ x_1 = -6 $, $ y_1 = 4 $:

$$
y - 4 = -\frac{1}{2}(x + 6)
$$

Distribute:

$$
y - 4 = -\frac{1}{2}x - 3
$$

Add 4 to both sides:

$$
y = -\frac{1}{2}x + 1
$$

✔ Answer: $ \boxed{y = -\frac{1}{2}x + 1} $

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6. Write the equation for a line that is parallel to the line $ 2x + 4y = 6 $ and passes through the point $ (4, 4) $.

Step 1: Rewrite $ 2x + 4y = 6 $ in slope-intercept form.

$$
4y = -2x + 6 \Rightarrow y = -\frac{1}{2}x + \frac{3}{2}
$$

So slope $ m = -\frac{1}{2} $

Step 2: Use point-slope form with $ m = -\frac{1}{2} $, point $ (4, 4) $:

$$
y - 4 = -\frac{1}{2}(x - 4)
$$

Distribute:

$$
y - 4 = -\frac{1}{2}x + 2
$$

Add 4:

$$
y = -\frac{1}{2}x + 6
$$

✔ Answer: $ \boxed{y = -\frac{1}{2}x + 6} $

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✔ Final Answers Summary:

1. Lines intersect; different slopes.
2. Same line; equations simplify to the same.
3. Parallel lines have same slope; same line has same slope and intercept.
4. $ \boxed{-2} $
5. $ \boxed{y = -\frac{1}{2}x + 1} $
6. $ \boxed{y = -\frac{1}{2}x + 6} $

Let me know if you'd like help graphing any of these or if the image had different numbers!