A straight line has a gradient of -2 and passes through the point (4,1). What is its equation?
Explanation
To find the equation of a straight line with a gradient (slope) of \(-2\) that passes through the point \((4, 1)\), we use the point-gradient form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
where:
- \(m\) is the gradient,
- \((x_1, y_1)\) is the point the line passes through.
For the given line:
- \(m = -2\),
- \((x_1, y_1) = (4, 1)\).
Substitute these values into the point-gradient form:
\[
y - 1 = -2(x - 4)
\]
Expand and simplify:
\[
y - 1 = -2x + 8
\]
\[
y = -2x + 8 + 1
\]
\[
y = -2x + 9
\]
To put this in the standard form \(Ax + By + C = 0\), rearrange:
\[
2x + y - 9 = 0
\]
Conclusion:
The correct equation of the line is C. \(y + 2x - 9 = 0\).