1. Write an equation of the line that passes through point P and is perpendicular to the line with the given equation P(1,4); Y= -2x +4
2. Write an equation of the line that passes through point P and is perpendicular to the line with the given equation P(5,3); Y= 5x +2
new slope = -1/old slope
so for the first one
y = +(1/2) x + b
now put in 1 for x and 4 for y to find b
Do the second one the same way
The equation for number 1. would be
Y= 1/2 x + 3.5?
The equation for 2. would be
Y= - 1/5 x + 4?
yes, although I would write #1 as 2y = x + 7
and
yes again but again I would write 5y = -x + 20
I don’t understand why do you wirte it like that?
To find the equation of a line that is perpendicular to a given line and passes through a given point, you need to use the concept of slope.
1. Let's start with the first question. The given equation is Y = -2x + 4, and the point is P(1,4).
a. First, determine the slope of the given line by looking at the coefficient of x. In this case, the slope is -2.
b. The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, the slope of the perpendicular line is 1/2.
c. Now that we have the slope, substitute the point P(1,4) into the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept.
d. Using the point-slope form, substitute the values: 4 = (1/2)(1) + b.
e. Simplify the equation: 4 = 1/2 + b.
f. Subtract 1/2 from both sides: 4 - 1/2 = b.
g. Find the common denominator: 8/2 - 1/2 = b.
h. Simplify: 7/2 = b.
i. The equation of the line that passes through point P(1,4) and is perpendicular to Y = -2x + 4 is y = (1/2)x + 7/2.
2. Now, let's move on to the second question. The given equation is Y = 5x + 2, and the point is P(5,3).
a. Find the slope of the given line by looking at the coefficient of x. In this case, the slope is 5.
b. The slope of the perpendicular line is the negative reciprocal of the original line's slope. So, the slope of the perpendicular line is -1/5.
c. Substitute the point P(5,3) into the slope-intercept form, y = mx + b.
d. Using the point-slope form, substitute the values: 3 = (-1/5)(5) + b.
e. Simplify the equation: 3 = -1 + b.
f. Add 1 to both sides: 3 + 1 = b.
g. The equation of the line that passes through point P(5,3) and is perpendicular to Y = 5x + 2 is y = (-1/5)x + 4.