Use point-slope form to write the equation of a line that passes through the point left parenthesis, 5, comma, 19, right parenthesis(5,19) with slope minus, start fraction, 2, divided by, 3, end fraction−32​

To write the equation of a line using the **point-slope form**, we use the formula:

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

Where:
- \( (x_1, y_1) \) is a point on the line,
- \( m \) is the slope of the line.

In this case, the given point is \( (5, 19) \) and the slope \( m = -\frac{2}{3} \).

Substitute these values into the point-slope form:

\[
y - 19 = -\frac{2}{3}(x - 5)
\]

This is the equation of the line in point-slope form. If you need it in another form, like slope-intercept form, we can simplify further:

1. Distribute the slope on the right-hand side:

\[
y - 19 = -\frac{2}{3}x + \frac{10}{3}
\]

2. Add 19 to both sides to solve for \( y \):

\[
y = -\frac{2}{3}x + \frac{10}{3} + 19
\]

3. Combine like terms:

\[
y = -\frac{2}{3}x + \frac{10}{3} + \frac{57}{3}
\]

\[
y = -\frac{2}{3}x + \frac{67}{3}
\]

So, the equation of the line in slope-intercept form is:

\[
y = -\frac{2}{3}x + \frac{67}{3}
\]

Would you like further details or explanations?

Here are 5 related questions:
1. How do you convert from point-slope form to slope-intercept form?
2. What is the slope-intercept form of a line, and how does it differ from point-slope form?
3. How do you find the slope of a line given two points?
4. Can we derive the equation of a line from the general form \(Ax + By = C\)?
5. What happens if the slope is zero or undefined when writing the equation of a line?

**Tip:** Point-slope form is especially useful when you know a point on the line and the slope but don't want to rearrange terms right away.

Use point-slope form to write the equation of a line that passes through the point (5,19) with slope -2/3.

Learn how to write the equation of a line using point-slope form, passing through the point (5,19) with a slope of -2/3. This detailed guide covers key algebraic concepts such as converting to slope-intercept form, suitable for students in Grades 8-10.

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