Question

left parenthesis, 0, comma, 4, right parenthesis(0,4) and left parenthesis, minus, 5, comma, 3, right parenthesis(−5,3) and determine the equation of the line.

Answer 1

To find the equation of the line that passes through the points \( (0, 4) \) and \( (-5, 3) \), 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 \((x_1, y_1) = (0, 4)\) and \((x_2, y_2) = (-5, 3)\).

Plugging in the values:
\[
m = \frac{3 - 4}{-5 - 0} = \frac{-1}{-5} = \frac{1}{5}
\]

2. **Use the point-slope form** of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the point \((0, 4)\) and the slope \(m = \frac{1}{5}\):
\[
y - 4 = \frac{1}{5}(x - 0)
\]
Simplifying this gives:
\[
y - 4 = \frac{1}{5}x
\]
\[
y = \frac{1}{5}x + 4
\]

3. **Convert to standard form** \(Ax + By = C\) if needed:
Multiplying the entire equation by 5 to eliminate the fraction:
\[
5y = x + 20
\]
Rearranging:
\[
-x + 5y = 20
\]
Or, in standard form:
\[
x - 5y = -20
\]

Thus, the equation of the line passing through the points \((0, 4)\) and \((-5, 3)\) is:
\[
y = \frac{1}{5}x + 4 \quad \text{or} \quad x - 5y = -20
\]