What is an equation of the line that passes through the point (6,-8) and is perpendicular to the line 2x-y=7

To find the equation of the line that passes through the point $(6,-8)$ and is perpendicular to the line $2x-y=7$, we need to follow these steps:<br /><br />1. Find the slope of the given line $2x-y=7$.<br />2. Find the slope of the line perpendicular to the given line.<br />3. Use the point-slope form of a linear equation to find the equation of the perpendicular line.<br /><br />Step 1: Find the slope of the given line $2x-y=7$.<br /><br />The given line can be rewritten in slope-intercept form as $y = 2x - 7$. The slope of this line is $2$.<br /><br />Step 2: Find the slope of the line perpendicular to the given line.<br /><br />The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope of the line perpendicular to the given line is $-\frac{1}{2}$.<br /><br />Step 3: Use the point-slope form of a linear equation to find the equation of the perpendicular line.<br /><br />The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.<br /><br />Substituting the point $(6,-8)$ and the slope $-\frac{1}{2}$ into the point-slope form, we get:<br /><br />$y - (-8) = -\frac{1}{2}(x - 6)$<br /><br />Simplifying this equation, we get:<br /><br />$y + 8 = -\frac{1}{2}x + 3$<br /><br />$y = -\frac{1}{2}x - 5$<br /><br />Therefore, the equation of the line that passes through the point $(6,-8)$ and is perpendicular to the line $2x-y=7$ is $y = -\frac{1}{2}x - 5$.