Question

. A line passes through the points (-2.7, 0.78) and (21, 15). Write the equation of the line in standard form. Show your work.

Answer 1

To find the equation of the line passing through the points \((-2.7, 0.78)\) and \((21, 15)\), we will follow these steps:

1. **Calculate the slope (m)**:
The slope \( m \) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \( (x_1, y_1) = (-2.7, 0.78) \) and \( (x_2, y_2) = (21, 15) \).

Substituting the values, we have:
\[
m = \frac{15 - 0.78}{21 - (-2.7)} = \frac{15 - 0.78}{21 + 2.7} = \frac{14.22}{23.7}
\]

Now we calculate \( \frac{14.22}{23.7} \):
\[
m \approx 0.599 \text{ (approximating to three decimal places)}
\]

2. **Use the point-slope form of the equation of a line**:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]

We can use point \((-2.7, 0.78)\):
\[
y - 0.78 = 0.599(x + 2.7)
\]

Now expanding this equation:
\[
y - 0.78 = 0.599x + 0.599 \cdot 2.7
\]
Calculate \( 0.599 \cdot 2.7 \):
\[
0.599 \cdot 2.7 \approx 1.6133
\]
Thus, we have:
\[
y - 0.78 = 0.599x + 1.6133
\]
Adding \( 0.78 \) to both sides:
\[
y = 0.599x + 1.6133 + 0.78
\]
\[
y = 0.599x + 2.3933
\]

3. **Convert to standard form \( Ax + By = C \)**:
We start from the slope-intercept form:
\[
y = 0.599x + 2.3933
\]
Rearranging gives:
\[
-0.599x + y = 2.3933
\]
To convert to standard form (preferably with integer coefficients), we multiply through by 1000 to eliminate the decimal:
\[
-599x + 1000y = 2393.3
\]

We can multiply everything by 10 to make it easier:
\[
-5990x + 10000y = 23933
\]

To make the leading coefficient positive \( A = 5990 \):
\[
5990x - 10000y = -23933
\]

Therefore, the equation of the line in standard form is:
\[
5990x - 10000y = -23933
\]

To summarize, the final answer in standard form is:
\[
5990x - 10000y = -23933
\]