Linear Equations Worksheets with Answer Key - Free Printable

Let's solve this step by step. The goal is to convert equations from slope-intercept form ($ y = mx + b $) to standard form ($ Ax + By + C = 0 $), where $ A $, $ B $, and $ C $ are integers, and $ A $ is non-negative.

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🔷 Part 1: Convert slope-intercept form to standard form

We start with equations in the form $ y = mx + b $, and we want to rearrange them into $ Ax + By + C = 0 $, with integer coefficients and $ A \geq 0 $.

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#### 1. $ y = -2x + 1 $

Start with:
$$
y = -2x + 1
$$

Move all terms to one side:
$$
2x + y - 1 = 0
$$

✔ Standard form:
$$
\boxed{2x + y - 1 = 0}
$$

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#### 2. $ y = -3x + 2 $

$$
y = -3x + 2
$$

Bring all terms to left:
$$
3x + y - 2 = 0
$$

✔ Standard form:
$$
\boxed{3x + y - 2 = 0}
$$

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#### 3. $ y = \frac{1}{4}x + \frac{3}{4} $

Eliminate fractions by multiplying every term by 4 (LCM of denominators):

$$
4y = x + 3
$$

Now move all terms to one side:
$$
-x + 4y - 3 = 0
$$

Make $ A $ positive by multiplying entire equation by $-1$:
$$
x - 4y + 3 = 0
$$

✔ Standard form:
$$
\boxed{x - 4y + 3 = 0}
$$

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#### 4. $ y = \frac{2}{7}x + \frac{1}{7} $

Multiply every term by 7:

$$
7y = 2x + 1
$$

Bring all terms to left:
$$
-2x + 7y - 1 = 0
$$

Multiply by $-1$ to make $ A $ positive:
$$
2x - 7y + 1 = 0
$$

✔ Standard form:
$$
\boxed{2x - 7y + 1 = 0}
$$

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#### 5. $ y = -6x - 2 $

$$
y = -6x - 2
$$

Bring all terms to left:
$$
6x + y + 2 = 0
$$

✔ Standard form:
$$
\boxed{6x + y + 2 = 0}
$$

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#### 6. $ y = -\frac{1}{6}x - \frac{5}{6} $

Multiply every term by 6:

$$
6y = -x - 5
$$

Bring all terms to left:
$$
x + 6y + 5 = 0
$$

✔ Standard form:
$$
\boxed{x + 6y + 5 = 0}
$$

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🔷 Part 2: Write equations in standard form using slope and y-intercept

We use the slope-intercept form first: $ y = mx + b $, then convert to standard form.

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#### 1. Slope = -8, y-intercept = 5

Equation:
$$
y = -8x + 5
$$

Bring to standard form:
$$
8x + y - 5 = 0
$$

✔ Standard form:
$$
\boxed{8x + y - 5 = 0}
$$

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#### 2. Slope = $ \frac{5}{2} $, y-intercept = 1

Equation:
$$
y = \frac{5}{2}x + 1
$$

Multiply every term by 2 to eliminate denominator:
$$
2y = 5x + 2
$$

Bring all terms to left:
$$
-5x + 2y - 2 = 0
$$

Multiply by $-1$ to make $ A $ positive:
$$
5x - 2y + 2 = 0
$$

✔ Standard form:
$$
\boxed{5x - 2y + 2 = 0}
$$

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#### 3. Slope = $ \frac{7}{2} $, y-intercept = 3

Equation:
$$
y = \frac{7}{2}x + 3
$$

Multiply by 2:
$$
2y = 7x + 6
$$

Bring to left:
$$
-7x + 2y - 6 = 0
$$

Multiply by $-1$:
$$
7x - 2y + 6 = 0
$$

✔ Standard form:
$$
\boxed{7x - 2y + 6 = 0}
$$

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#### 4. Slope = $ \frac{3}{5} $, y-intercept = 1

Equation:
$$
y = \frac{3}{5}x + 1
$$

Multiply by 5:
$$
5y = 3x + 5
$$

Bring to left:
$$
-3x + 5y - 5 = 0
$$

Multiply by $-1$:
$$
3x - 5y + 5 = 0
$$

✔ Standard form:
$$
\boxed{3x - 5y + 5 = 0}
$$

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✔ Final Answers Summary:

#### Part 1: Converting given equations

1. $ 2x + y - 1 = 0 $
2. $ 3x + y - 2 = 0 $
3. $ x - 4y + 3 = 0 $
4. $ 2x - 7y + 1 = 0 $
5. $ 6x + y + 2 = 0 $
6. $ x + 6y + 5 = 0 $

#### Part 2: From slope and y-intercept

1. $ 8x + y - 5 = 0 $
2. $ 5x - 2y + 2 = 0 $
3. $ 7x - 2y + 6 = 0 $
4. $ 3x - 5y + 5 = 0 $

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Let me know if you'd like these written in a different format (like $ Ax + By = C $ instead of $ Ax + By + C = 0 $).