Algebra 2 Practice - Change Point-Slope Form to Standard Form

### Algebra 2 Practice: Changing Point-Slope Form to Standard Form

When working with linear equations, one common task is converting from **point-slope form** to **standard form**. This skill is useful for rewriting equations in a format that is often easier to interpret or use in applications like graphing or solving systems of equations.

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### Forms of Linear Equations:

1. **Point-Slope Form**:
\[
y - y_1 = m(x - x_1)
\]
Where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.

2. **Standard Form**:
\[
Ax + By = C
\]
Where \(A\), \(B\), and \(C\) are integers, and \(A\) is typically positive.

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### Steps to Convert:

1. **Expand Point-Slope Form**: Distribute \(m\) to simplify the equation.
2. **Rearrange Terms**: Move all terms involving \(x\) and \(y\) to one side of the equation and the constant to the other.
3. **Eliminate Fractions (if any)**: Multiply through by the least common denominator to make all coefficients integers.
4. **Adjust Signs (if needed)**: Ensure \(A\) is positive to match the standard form convention.

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### Example Problem:
Convert the equation \(y - 3 = \frac{2}{3}(x - 6)\) to standard form.

#### Step-by-Step Solution:
1. **Start with Point-Slope Form**:
\(y - 3 = \frac{2}{3}(x - 6)\).

2. **Distribute the Slope**:
\(y - 3 = \frac{2}{3}x - 4\).

3. **Eliminate Fractions**: Multiply the entire equation by 3:
\(3(y - 3) = 2x - 12\).
\(3y - 9 = 2x - 12\).

4. **Rearrange Terms**: Move \(x\) and \(y\) terms to one side:
\(-2x + 3y = -3\).

5. **Adjust Signs**: Multiply through by \(-1\) to make \(A\) positive:
\(2x - 3y = 3\).

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### Final Answer:
The standard form of the equation is:
\[
2x - 3y = 3
\]

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### Practice Problems:
1. Convert \(y + 2 = -4(x - 1)\) to standard form.
2. Rewrite \(y - 5 = \frac{1}{2}(x + 8)\) in standard form.
3. Change \(y - 7 = 3(x - 4)\) into standard form.

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### Key Tips:
- Always simplify fractions early to avoid errors.
- Remember the sign convention for standard form.
- Double-check your work by substituting a point to verify the equation matches the original.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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