Geometry Worksheets | Lines Worksheets - Free Printable

To solve each problem, we need to find the equation of a line that:

1. Passes through a given point, and
2. Is parallel to a given line.

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🔑 Key Concept:

> Parallel lines have the same slope.

So, our strategy is:

- Step 1: Find the slope of the given line.
- Step 2: Use the point-slope form of a line:
`y - y₁ = m(x - x₁)`
where `(x₁, y₁)` is the given point and `m` is the slope from Step 1.
- Step 3: Rearrange into slope-intercept form: `y = mx + b`.

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Let’s solve each problem one by one.

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## ✔ Problem 1: Point (-1, -1), Line: -2x + 3y = 21

Step 1: Find slope of given line

Rewrite in slope-intercept form:

```
-2x + 3y = 21
→ 3y = 2x + 21
→ y = (2/3)x + 7
```

✔ Slope `m = 2/3`

Step 2: Use point-slope form with point (-1, -1)

```
y - (-1) = (2/3)(x - (-1))
→ y + 1 = (2/3)(x + 1)
→ y = (2/3)x + 2/3 - 1
→ y = (2/3)x - 1/3
```

✔ Answer: y = (2/3)x - 1/3

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## ✔ Problem 2: Point (-1, 1), Line: y = -4/9 x - 3

Step 1: Slope is already given → m = -4/9

Step 2: Use point-slope form

```
y - 1 = (-4/9)(x - (-1))
→ y - 1 = (-4/9)(x + 1)
→ y = (-4/9)x - 4/9 + 1
→ y = (-4/9)x + 5/9
```

✔ Answer: y = (-4/9)x + 5/9

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## ✔ Problem 3: Point (2, 5), Line: x + 3y = 3

Step 1: Solve for y

```
x + 3y = 3
→ 3y = -x + 3
→ y = (-1/3)x + 1
```

✔ Slope `m = -1/3`

Step 2: Point-slope form

```
y - 5 = (-1/3)(x - 2)
→ y = (-1/3)x + 2/3 + 5
→ y = (-1/3)x + 17/3
```

✔ Answer: y = (-1/3)x + 17/3

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## ✔ Problem 4: Point (1, -4), Line: y = 3x - 4

Step 1: Slope m = 3

Step 2: Point-slope

```
y - (-4) = 3(x - 1)
→ y + 4 = 3x - 3
→ y = 3x - 7
```

✔ Answer: y = 3x - 7

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## ✔ Problem 5: Point (-3, -2), Line: y = (3/2)x + 3

Step 1: Slope m = 3/2

Step 2: Point-slope

```
y - (-2) = (3/2)(x - (-3))
→ y + 2 = (3/2)(x + 3)
→ y = (3/2)x + 9/2 - 2
→ y = (3/2)x + 5/2
```

✔ Answer: y = (3/2)x + 5/2

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## ✔ Problem 6: Point (-2, 0), Line: 6x - y = 1

Step 1: Solve for y

```
6x - y = 1
→ -y = -6x + 1
→ y = 6x - 1
```

✔ Slope `m = 6`

Step 2: Point-slope

```
y - 0 = 6(x - (-2))
→ y = 6(x + 2)
→ y = 6x + 12
```

✔ Answer: y = 6x + 12

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## ✔ Problem 7: Point (2, 2), Line: 2x + 9y = 18

Step 1: Solve for y

```
2x + 9y = 18
→ 9y = -2x + 18
→ y = (-2/9)x + 2
```

✔ Slope `m = -2/9`

Step 2: Point-slope

```
y - 2 = (-2/9)(x - 2)
→ y = (-2/9)x + 4/9 + 2
→ y = (-2/9)x + 22/9
```

✔ Answer: y = (-2/9)x + 22/9

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## ✔ Problem 8: Point (-3, -2), Line: y = (-3/4)x + 4

Step 1: Slope m = -3/4

Step 2: Point-slope

```
y - (-2) = (-3/4)(x - (-3))
→ y + 2 = (-3/4)(x + 3)
→ y = (-3/4)x - 9/4 - 2
→ y = (-3/4)x - 17/4
```

✔ Answer: y = (-3/4)x - 17/4

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## 📝 Final Answers Summary:

1. y = (2/3)x - 1/3
2. y = (-4/9)x + 5/9
3. y = (-1/3)x + 17/3
4. y = 3x - 7
5. y = (3/2)x + 5/2
6. y = 6x + 12
7. y = (-2/9)x + 22/9
8. y = (-3/4)x - 17/4

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✔ All answers are in slope-intercept form (y = mx + b) as requested.

Let me know if you’d like these graphed or explained differently!