To find the equation of line \( t \) that passes through the point \( (4, 5) \) and is perpendicular to the given line, we first need to determine the slope of the given line.
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Find the slope of the given line: The line passes through the points \( (0, 3) \) and \( (5, 2) \). We can calculate the slope \( m \) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 3}{5 - 0} = \frac{-1}{5} \]
So, the slope of the given line is \( -\frac{1}{5} \).
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Find the perpendicular slope: The slope of line \( t \), which is perpendicular to the given line, is the negative reciprocal of \( -\frac{1}{5} \):
\[ m_t = 5 \]
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Write the equation of line \( t \): Now, we can use the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Using point \( (4, 5) \) and slope \( 5 \):
\[ y - 5 = 5(x - 4) \]
Simplifying this:
\[ y - 5 = 5x - 20 \] \[ y = 5x - 15 \]
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Convert to standard form: The standard form of a linear equation is \( Ax + By = C \). To convert \( y = 5x - 15 \):
\[ -5x + y = -15 \]
Multiplying by -1 to make the coefficients positive:
\[ 5x - y = 15 \]
So, in standard form, the equation of line \( t \) is:
\[ 5x - y = 15 \]
Thus, the correct answer is:
\[ \boxed{5x - y = 15} \]
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