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The point-slope form is a useful tool when you know one point on a line and the slope of that line. It provides a straightforward way to express the equation of a line, making it an excellent starting point for further conversion into other formats. Generally, the formula looks like this:

\[ y - y_1 = m(x - x_1) \]

Where:

• \( x_1, y_1 \) are the coordinates of the given point.
• \( m \) is the slope of the line.
• \( x \) and \( y \) are the variables.

In our exercise, we used the point \((-4,0)\) and the slope \(-\frac{1}{10}\) to write the equation in point-slope form. We substituted these into the equation like this:

\[ y - 0 = -\frac{1}{10}(x + 4) \]

Then, we simplified it to get:

\[ y = -\frac{1}{10}x - \frac{2}{5} \]

This form is particularly helpful when focusing on the relationship between slope and line orientation.

The standard form of a linear equation is another fundamental way to describe a line algebraically. It typically appears as:

\[ Ax + By = C \]

Here, \( A \), \( B \), and \( C \) are integers and \( A \) should be non-negative. The equation helps clearly express a line without fractional coefficients, which can be advantageous for certain algebraic operations or integrations.

• To convert the equation \( y = -\frac{1}{10}x - \frac{2}{5} \) into standard form, multiply through by 10 to clear the fraction, obtaining: \( 10y = -x - 4 \).
• Then, rearrange and simplify it to get \( x + 10y = -4 \).

This equation now satisfies the standard format and is confirmed by checking that both original points, \((-4,0)\) and \(6,-1)\), validate it when substituted back.

Calculating the slope of a line is a fundamental concept in understanding linear functions. The slope \( m \) represents the tangent of the angle that a line makes with the positive x-axis. It indicates how steep the line is, and whether it is ascending, descending, or horizontal. To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula calculates the 'rise over run' or the change in y over the change in x.

• Substituting the points \((-4, 0)\) and \((6, -1)\), we get: \( \frac{-1 - 0}{6 - (-4)} = \frac{-1}{10} \).
• This gives a negative slope, indicating the line descends from left to right.

Understanding how to calculate and interpret the slope is crucial for graphing lines and understanding their behavior relative to the x and y axes.