To find the equation of a line in standard form that passes through the point \((-4, 2)\) with a slope of \(\frac{9}{2}\), we can start with the point-slope form of the equation:
\[ y - y_1 = m(x - x_1) \]
Here, \(m\) is the slope, and \((x_1, y_1)\) is the point the line passes through. Substituting in our values:
\[ y - 2 = \frac{9}{2}(x + 4) \]
Now, let's simplify this equation:
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Expand the right side: \[ y - 2 = \frac{9}{2}x + \frac{9}{2} \cdot 4 \] \[ y - 2 = \frac{9}{2}x + 18 \]
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Add 2 to both sides to isolate \(y\): \[ y = \frac{9}{2}x + 18 + 2 \] \[ y = \frac{9}{2}x + 20 \]
Now, we convert this into standard form \(Ax + By = C\). We start by multiplying the entire equation by 2 to eliminate the fraction:
\[ 2y = 9x + 40 \]
Next, we rearrange this equation to get it into standard form:
\[ -9x + 2y = 40 \]
To conform to the typical standard form \(Ax + By = C\), we can multiply by -1 to make the coefficient of \(x\) positive:
\[ 9x - 2y = -40 \]
Now we can match this with the options given:
- \(9x - 2y = 36\) (Not a match)
- \(9x - 2y = 26\) (Not a match)
- \(9x - 2y = -40\) (This matches)
- \(9x - 2y = -10\) (Not a match)
Thus, the correct equation in standard form that passes through the point (-4, 2) and has a slope of \(\frac{9}{2}\) is:
C. \(9x - 2y = -40\)
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