Question

equation of a line passing through two points C(5, 6) and D(8, 11)

Answer 1

To find the equation of the line passing through the points \( C(5, 6) \) and \( D(8, 11) \), we can follow these steps:

1. **Calculate the slope (m)** of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \( (x_1, y_1) = (5, 6) \) and \( (x_2, y_2) = (8, 11) \).

Substituting in the values:
\[
m = \frac{11 - 6}{8 - 5} = \frac{5}{3}
\]

2. **Use the point-slope form of the equation of a line**:
\[
y - y_1 = m(x - x_1)
\]
Using point \( C(5, 6) \) and the slope \( \frac{5}{3} \):
\[
y - 6 = \frac{5}{3}(x - 5)
\]

3. **Simplify to get the slope-intercept form (y = mx + b)**:
Distributing the slope on the right side:
\[
y - 6 = \frac{5}{3}x - \frac{25}{3}
\]

Adding 6 (or \( \frac{18}{3} \)) to both sides:
\[
y = \frac{5}{3}x - \frac{25}{3} + \frac{18}{3}
\]
\[
y = \frac{5}{3}x - \frac{7}{3}
\]

Thus, the equation of the line in slope-intercept form is:
\[
y = \frac{5}{3}x - \frac{7}{3}
\]

If you prefer the standard form \( Ax + By = C \), we can rearrange:
\[
5x - 3y = 7
\]

So the equation of the line can be expressed in either form:
1. Slope-intercept form: \( y = \frac{5}{3}x - \frac{7}{3} \)
2. Standard form: \( 5x - 3y = 7 \)