To write the equation of a line in standard form, we need to express it as Ax + By = C, where A, B, and C are constants. Given that the line passes through the point (7, -3) and has a y-intercept of 2, we can proceed as follows:
1. Determine the slope (m) of the line using the given information.
The slope of a line can be found using the formula:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) and (x2, y2) are two points on the line.
We have the point (7, -3) and the y-intercept (0, 2). Plugging these values into the formula:
m = (2 - (-3)) / (0 - 7)
= 5 / (-7)
= -5/7.
So, the slope of the line is -5/7.
2. Write the equation using the point-slope form.
The point-slope form of a line is:
y - y1 = m(x - x1),
where (x1, y1) is a point on the line, and m is the slope.
Using the values (7, -3) and m = -5/7:
y - (-3) = (-5/7)(x - 7)
y + 3 = (-5/7)(x - 7).
3. Convert the equation to standard form.
To convert the equation to standard form (Ax + By = C), we multiply through by 7 to clear the fraction:
7(y + 3) = -5(x - 7)
7y + 21 = -5x + 35.
Rearranging the terms:
5x + 7y = 14.
Therefore, the equation of the line in standard form that passes through the point (7, -3) and has a y-intercept of 2 is 5x + 7y = 14.