Find an equation for the linear function g(x) which is perpendicular to the line 5x-8y=32 and intersects the line 5x-8y=32 at x=32 g(x)=

To find the equation of the linear function $g(x)$ that is perpendicular to the line $5x - 8y = 32$ and intersects it at $x = 32$, we need to follow these steps:<br /><br />1. Find the slope of the given line $5x - 8y = 32$.<br />2. Determine the slope of the perpendicular line $g(x)$.<br />3. Find the point of intersection at $x = 32$.<br />4. Use the point-slope form to write the equation of $g(x)$.<br /><br />Step 1: Find the slope of the given line $5x - 8y = 32$.<br />To find the slope, we can rewrite the equation in slope-intercept form $y = mx + b$, where $m$ is the slope.<br /><br />$5x - 8y = 32$<br />$-8y = -5x + 32$<br />$y = \frac{5}{8}x - 4$<br /><br />The slope of the given line is $\frac{5}{8}$.<br /><br />Step 2: Determine the slope of the perpendicular line $g(x)$.<br />The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.<br /><br />The slope of the perpendicular line $g(x)$ is $-\frac{8}{5}$.<br /><br />Step 3: Find the point of intersection at $x = 32$.<br />Substitute $x = 32$ into the equation of the given line to find the corresponding $y$-coordinate.<br /><br />$5(32) - 8y = 32$<br />$160 - 8y = 32$<br />$-8y = -128$<br />$y = 16$<br /><br />So, the point of intersection is $(32, 16)$.<br /><br />Step 4: Use the point-slope form to write the equation of $g(x)$.<br />The point-slope form of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.<br /><br />Using the point $(32, 16)$ and the slope $-\frac{8}{5}$, we can write the equation of $g(x)$ as:<br /><br />$y - 16 = -\frac{8}{5}(x - 32)$<br /><br />Simplifying this equation, we get:<br /><br />$y - 16 = -\frac{8}{5}x + \frac{256}{5}$<br />$y = -\frac{8}{5}x + \frac{256}{5} + 16$<br />$y = -\frac{8}{5}x + \frac{256}{5} + \frac{80}{5}$<br />$y = -\frac{8}{5}x + \frac{336}{5}$<br /><br />Therefore, the equation for the linear function $g(x)$ is:<br /><br />$g(x) = -\frac{8}{5}x + \frac{336}{5}$