Question

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find the equation of the line perpendicular to $y = -\frac{1}{2}x - 5$ that passes through the point $(2, 7)$. write this line in slope - intercept form.
$y = $

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = -\frac{1}{2}x - 5 \) is \( m_1 = -\frac{1}{2} \). The slope of a line perpendicular to a line with slope \( m_1 \) is the negative reciprocal, so \( m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{2}} = 2 \).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(2,7) \) and \( m = 2 \). Substituting these values, we get \( y - 7 = 2(x - 2) \).

Step3: Convert to slope - intercept form

Expand the right - hand side: \( y - 7 = 2x-4 \). Then add 7 to both sides: \( y=2x - 4 + 7 \), so \( y = 2x+3 \).

Answer:

\( 2x + 3 \)