4) The equation of line p is y=-4x+1 Line q is perpendicular to line p and passes through (-6,1) . What is the equation of line q?

To find the equation of line q, we need to determine its slope and use the point-slope form of a linear equation.<br /><br />Step 1: Find the slope of line p.<br />The equation of line p is given as $y=-4x+1$. The slope of a line in the form $y=mx+b$ is the coefficient of x. Therefore, the slope of line p is -4.<br /><br />Step 2: Determine the slope of line q.<br />Since line q is perpendicular to line p, its slope will be the negative reciprocal of the slope of line p. The negative reciprocal of -4 is $\frac{1}{4}$.<br /><br />Step 3: Use the point-slope form to find the equation of line q.<br />The point-slope form of a linear equation 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. We are given that line q passes through the point $(-6,1)$, so we can use this point and the slope we found in step 2.<br /><br />Substituting the values into the point-slope form, we have:<br />$y-1=\frac{1}{4}(x-(-6))$<br /><br />Simplifying further:<br />$y-1=\frac{1}{4}(x+6)$<br /><br />Expanding the right side:<br />$y-1=\frac{1}{4}x+\frac{6}{4}$<br /><br />Simplifying the fraction:<br />$y-1=\frac{1}{4}x+\frac{3}{2}$<br /><br />Adding 1 to both sides:<br />$y=\frac{1}{4}x+\frac{3}{2}+1$<br /><br />Combining like terms:<br />$y=\frac{1}{4}x+\frac{5}{2}$<br /><br />Therefore, the equation of line q is $y=\frac{1}{4}x+\frac{5}{2}$.