Example Question - equation of perpendicular line
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
Finding Equation of Perpendicular Line through Point
To find the equation of the line h, which is perpendicular to line g and passes through the point (3, -9), we need to start by identifying the slope of line g. The equation for line g is written in slope-intercept form as: \[ y = \frac{3}{10}x - 8 \] The slope (m) of line g is the coefficient of x, which is \(\frac{3}{10}\). For a line to be perpendicular to another line, its slope must be the negative reciprocal of the original line's slope. The negative reciprocal of \(\frac{3}{10}\) is \(-\frac{10}{3}\) (because when you flip the fraction and change the sign, you get the negative reciprocal). Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line to find line h's equation. Point-slope form is: \[ y - y_1 = m(x - x_1) \] where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Using the slope of line h, which is \(-\frac{10}{3}\), and the given point (3, -9), we plug these into the point-slope form: \[ y - (-9) = -\frac{10}{3}(x - 3) \] Simplify the equation: \[ y + 9 = -\frac{10}{3}x + \frac{10}{3} \cdot 3 \] \[ y + 9 = -\frac{10}{3}x + 10 \] We want to get it into slope-intercept form (y = mx + b), so we'll isolate y: \[ y = -\frac{10}{3}x + 10 - 9 \] \[ y = -\frac{10}{3}x + 1 \] The equation of line h in slope-intercept form, with numbers as simplified proper fractions, improper fractions, or integers, is: \[ y = -\frac{10}{3}x + 1 \]
Finding Equation of Perpendicular Line
To find the equation of line h, which is perpendicular to line g, we will follow these steps: 1. Identify the slope of line g. 2. Determine the slope of line h. 3. Use the point-slope form to create the equation of line h. 4. Convert the equation into slope-intercept form. The equation for line g is given as \( y = \frac{3}{10}x - 8 \). The slope of line g is the coefficient of x, which is \( \frac{3}{10} \). Lines that are perpendicular to each other have slopes that are negative reciprocals. Therefore, if the slope of line g is \( \frac{3}{10} \), the slope of line h will be its negative reciprocal, which is \( -\frac{10}{3} \). Now, we know that line h has a slope of \( -\frac{10}{3} \) and it passes through the point (3, -9). We can use the point-slope form of the equation to find the equation of line h: \( y - y_1 = m(x - x_1) \) Substitute m (slope) with \( -\frac{10}{3} \) and \( (x_1, y_1) \) with (3, -9): \( y - (-9) = -\frac{10}{3}(x - 3) \) Simplify and solve for y to get the equation in slope-intercept form: \( y + 9 = -\frac{10}{3}x + 10 \) \( y = -\frac{10}{3}x + 10 - 9 \) \( y = -\frac{10}{3}x + 1 \) Therefore, the equation of line h in slope-intercept form is \( y = -\frac{10}{3}x + 1 \), with the numbers in the equation as simplified improper fractions or integers.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com