This worksheet helps students practice graphing lines and understanding the relationship between parallel lines and their 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 algebra worksheet involving graphing lines, analyzing their relationships, and writing equations of parallel lines.

Let me solve each problem step by step as if I had the image, and explain the reasoning clearly.

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1. Graph both lines on the same coordinate plane:

a) y = 2x – 3
b) y = –3x + 1

#### Step-by-step solution:

- For y = 2x – 3:
- This is in slope-intercept form: $ y = mx + b $
- Slope $ m = 2 $, y-intercept $ b = -3 $
- Plot the point (0, -3), then use rise/run: up 2, right 1 to get another point (1, -1), etc.

- For y = –3x + 1:
- Slope $ m = -3 $, y-intercept $ b = 1 $
- Plot (0, 1), then down 3, right 1 → (1, -2)

#### What do you notice about the two lines?
- They have different slopes (2 and -3)
- They are not parallel — they intersect at some point
- One has positive slope (rising), one has negative slope (falling)

#### What do you notice about the equations?
- Both are linear equations in slope-intercept form.
- Different slopes → not parallel.
- Different y-intercepts.

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2. Graph both lines on the same coordinate plane:

a) y = x – 2
b) y = x + 4

#### Step-by-step solution:

- y = x – 2: slope = 1, y-intercept = -2 → plot (0, -2), then (1, -1), (2, 0), etc.
- y = x + 4: slope = 1, y-intercept = 4 → plot (0, 4), (1, 5), etc.

#### What do you notice about the two lines?
- Both lines have the same slope (1) → they are parallel
- They never intersect
- They are shifted vertically relative to each other

#### What do you notice about the equations?
- Same slope coefficient (1), different y-intercepts (-2 and 4)
- This confirms they are parallel lines — same slope, different intercepts

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3. Draw a conclusion that summarizes what you observed in complete sentences.

> Lines with the same slope are parallel and never intersect. Lines with different slopes will intersect at some point. The y-intercept determines where the line crosses the y-axis, but does not affect whether lines are parallel.

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

- The given line is $ y = -2x $
- Its slope is $ -2 $
- Parallel lines have equal slopes

✔ So, the slope of any line parallel to this one is –2

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5. Write the equation for a line that is parallel to the line y = 5 – 1/2 x and passes through the point (–2, 4)

First, rewrite the given line in slope-intercept form:

$ y = 5 - \frac{1}{2}x $ → $ y = -\frac{1}{2}x + 5 $

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

We want a line with the same slope, passing through $ (-2, 4) $

Use point-slope form:
$$
y - y_1 = m(x - x_1)
$$
Plug in:
$$
y - 4 = -\frac{1}{2}(x + 2)
$$

Now simplify:
$$
y - 4 = -\frac{1}{2}x - 1 \\
y = -\frac{1}{2}x + 3
$$

✔ Final answer: $ \boxed{y = -\frac{1}{2}x + 3} $

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6. Write the equation for a line that is parallel to the line 2x + 3y = 6 and passes through the point (3, 4)

#### Step 1: Find the slope of the given line

Start with: $ 2x + 3y = 6 $

Solve for y:
$$
3y = -2x + 6 \\
y = -\frac{2}{3}x + 2
$$

So, slope $ m = -\frac{2}{3} $

#### Step 2: Use point-slope form with point (3, 4)

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

Simplify:
$$
y - 4 = -\frac{2}{3}x + 2 \\
y = -\frac{2}{3}x + 6
$$

✔ Final answer: $ \boxed{y = -\frac{2}{3}x + 6} $

(Alternatively, you could write in standard form: multiply by 3: $ 3y = -2x + 18 $ → $ 2x + 3y = 18 $)

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✔ Summary of Answers:

1. Lines: one rising (slope 2), one falling (slope -3); different slopes → intersect
2. Both have slope 1 → parallel; same slope, different intercepts
3. Conclusion: Lines with equal slopes are parallel; different slopes mean they intersect.
4. Slope of parallel line to $ y = -2x $: –2
5. Equation: $ \boxed{y = -\frac{1}{2}x + 3} $
6. Equation: $ \boxed{y = -\frac{2}{3}x + 6} $ or $ \boxed{2x + 3y = 18} $

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