How and why you convert -7x+6y to +7x-6y? 

When we convert \(-7x + 6y = -42\) to \(7x - 6y + 42 = 0\), we are essentially rearranging and normalizing the equation to a standard form. Here’s the step-by-step reasoning:

1. **Starting with the given equation**:
   \[
   -7x + 6y = -42
   \]

2. **Move all terms to one side of the equation**:
   To transform the equation into the standard form \(Ax + By + C = 0\), we can add \(7x\) and subtract \(6y\) from both sides:
   \[
   -7x + 7x + 6y - 6y = -42 + 7x - 6y
   \]
   Simplifying this, we get:
   \[
   0 = 7x - 6y - 42
   \]

3. **Rewriting the equation**:
   It is common to write the equation with the left-hand side (LHS) containing all the variables and constants, and zero on the right-hand side (RHS). Thus, we write:
   \[
   7x - 6y + 42 = 0
   \]

This rearrangement is done to match the standard form of a linear equation, which is typically written as \(Ax + By + C = 0\). The signs change because we are moving terms from one side of the equation to the other, effectively adding or subtracting terms to both sides to maintain equality.

To summarize, we converted \(-7x + 6y = -42\) to \(7x - 6y + 42 = 0\) to present the equation in its standard form, making it easier to read and compare with other linear equations.

Would you like further details or have any questions about this solution?

Here are 5 more questions you can ask:
1. How do you find the slope-intercept form of a linear equation from the standard form?
2. What is the difference between the standard form and the slope-intercept form of a linear equation?
3. How do you determine the perpendicular bisector of a line segment given two points?
4. Can you explain how to complete the square for solving quadratic equations?
5. What are the steps to graphing a linear equation?

**Tip:** When rearranging equations, always check that the resulting equation represents the same relationship by plugging in a couple of points to verify the equality.

Explore how to convert -7x + 6y to 7x - 6y with a detailed explanation. Learn about rearranging equations, transforming equations to standard form, and the reasons behind sign changes. Suitable for high school students and beyond.

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