Example Question - perpendicular lines
Here are examples of questions we've helped users solve.
Equation of Perpendicular Line
The equation of line p is given as y = -4x + 1. This equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. For line p, the slope is -4. Line q is supposed to be perpendicular to line p. When two lines are perpendicular, the product of their slopes is -1. This means that if the slope of line p is m, then the slope of line q will be -1/m. Therefore, the slope of line q is -1/(-4) = 1/4. Now that we know the slope of line q is 1/4, we can use the point it passes through, (-6, 1), to find the y-intercept (b) of line q. Starting with the point-slope form of the line equation: y - y1 = m(x - x1) Plugging in the slope (m = 1/4) and the point (-6, 1): y - 1 = 1/4(x - (-6)) y - 1 = 1/4(x + 6) Now, distribute 1/4 to (x + 6): y - 1 = 1/4x + 1/4(6) Simplify: y - 1 = 1/4x + 6/4 y - 1 = 1/4x + 3/2 Finally, we want to write this in slope-intercept form, so we solve for y by adding 1 to both sides: y = 1/4x + 3/2 + 1 Since we want to write numbers as simplified fractions or integers, let's convert 1 to a fraction with a denominator of 2: y = 1/4x + 3/2 + 2/2 Combine the fractions: y = 1/4x + 5/2 And so, the equation of line q in slope-intercept form is: y = 1/4x + 5/2
Equations of Lines and Points
The task is to find the equations of lines that are: (a) Parallel to the given line (b) Perpendicular to the given line Both lines must pass through a given point. The point in this instance is (-7, -2), and the given line is x = 1. (a) A line parallel to x = 1 is also a vertical line, because x = 1 is vertical. All vertical lines have equations of the form x = c, where c is a constant. Since the line we're trying to find must pass through the point (-7, -2), the value of c must be -7. Therefore, the equation of the line parallel to x = 1 and passing through (-7, -2) is: x = -7 (b) A line perpendicular to a vertical line is a horizontal line. All horizontal lines have equations of the form y = k, where k is a constant. Since the line we're trying to find must pass through the point (-7, -2), the value of k must be -2. Therefore, the equation of the line perpendicular to x = 1 and passing through (-7, -2) is: y = -2 So the answers are: (a) x = -7 (b) y = -2
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