To find the standard form of the equation of a line given a point and a slope, we start by using the point-slope form of the equation, which is given by:
y – y1 = m(x – x1)
Here, (x1, y1) is the point on the line (in this case, (1, 2)), and m is the slope (in this case, 7).
Substituting the values into the point-slope equation:
y - 2 = 7(x - 1)
Now, let’s simplify and rearrange this equation. First, distribute the slope:
y - 2 = 7x - 7
Next, add 2 to both sides to isolate y:
y = 7x - 7 + 2
This simplifies to:
y = 7x - 5
To convert this to standard form, which is written as Ax + By = C, we need to rearrange the equation:
-7x + y = -5
To arrange it further into the form where A is positive (as is customary), we can multiply the entire equation by -1:
7x – y = 5
Thus, the standard form of the equation of the line that passes through the point (1, 2) with a slope of 7 is:
7x – y = 5