To write an equation for a line that is perpendicular to another line, you need to follow these steps:
1. Find the slope of the given line.
2. Take the opposite reciprocal of that slope to find the slope of the perpendicular line.
3. Use the slope-intercept form of a line equation (y = mx + b), where m is the slope and b is the y-intercept, to write the equation of the perpendicular line.
Let's apply these steps to the problem you mentioned: Finding an equation of the line that contains (1,8) and is perpendicular to y = 3/4x + 1.
First, identify the slope of the given line. In this case, the slope is 3/4.
Next, find the slope of the perpendicular line by taking its opposite reciprocal. Inverting the slope gives us -4/3.
Now, plug the slope (-4/3) into the slope-intercept form of a line equation, along with the coordinates of the given point (1,8). Let's call the equation of the perpendicular line y = mx + b.
Plugging in the values, we have:
8 = (-4/3)(1) + b
To solve for b, multiply -4/3 by 1 and add it to both sides:
8 = -4/3 + b
To get rid of the fraction, rewrite 8 as 24/3:
24/3 = -4/3 + b
Next, simplify:
24/3 = -4/3 + 3b/3
Combine the fractions:
8 = -4 + 3b/3
To isolate b, subtract -4 from 8:
8 + 4 = 3b/3
12 = 3b/3
Again, to simplify fractions, multiply both sides by 3:
3 * 12 = 3 * 3b/3
36 = 3b
Divide both sides by 3:
36/3 = 3b/3
12 = b
So the y-intercept, b, is 12.
Now that we have the slope (-4/3) and the y-intercept (12), we can write the equation of the perpendicular line:
y = (-4/3)x + 12
This is the correct equation of the line that contains the point (1,8) and is perpendicular to y = 3/4x + 1.