Find an equation of the line with the slope 𝑚=19 that passes through the point (−6,2). Write the equation in the form 𝐴𝑥+𝐵𝑦=𝐶.


To find the equation of the line with a given slope \( m = 19 \) that passes through the point \((-6, 2)\), we can start by using the point-slope form of a line equation, which is:

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

Here, \((x_1, y_1)\) is the point \((-6, 2)\) and \(m = 19\). Substituting these values into the equation:

\[
y - 2 = 19(x + 6)
\]

Now, let's expand and simplify the equation:

\[
y - 2 = 19x + 114
\]

Next, add 2 to both sides to isolate \(y\):

\[
y = 19x + 116
\]

This equation is now in the slope-intercept form \(y = mx + b\). To convert it to the standard form \(Ax + By = C\), we'll rearrange the terms:

\[
-19x + y = 116
\]

To match the form \(Ax + By = C\), it's typical to have \(A\) as a positive integer. So, we can multiply the entire equation by \(-1\) to get:

\[
19x - y = -116
\]

Thus, the equation of the line in standard form is:

\[
\boxed{19x - y = -116}
\]

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How do you convert from point-slope form to standard form?
2. What are the differences between standard form and slope-intercept form?
3. How can you determine if two lines are parallel or perpendicular using their equations?
4. What is the geometric interpretation of the standard form equation of a line?
5. How do you find the x-intercept of a line given its equation in standard form?
6. Can you find the equation of a line perpendicular to this one that passes through a given point?
7. How do you derive the formula for the distance from a point to a line?
8. What are the applications of the standard form of a line equation in real-world problems?

**Tip:** When writing the equation of a line, always check whether the coefficients can be simplified for easier interpretation or if they can be converted into a preferred format.

Learn how to find the equation of a line with slope 19 passing through the point (-6, 2). Step-by-step solution provided with conversion from point-slope form to standard form. Suitable for Grades 7-9.

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