Find the equation for each of the following items below:

a. A line that passes through (6,26) and has a slope of 3.

b. A line that passes though the points (5,5) and (10,20)

c. A line that passes through (9,25) and has a slope of -3.

d. A line that passes through (1,1) and is perpendicular to the line 4x+6Y=18.

e. A circle with a radius of 4 and a midpoint at (2,2)

We will be happy to critique your thinking on this.

49 is 80% of

a. To find the equation of a line that passes through a given point (x1, y1) with a given slope (m), we can use the point-slope form of the equation:

y - y1 = m(x - x1)

In this case, the point (x1, y1) is (6, 26) and the slope (m) is 3. Plugging in these values into the point-slope form, we get:

y - 26 = 3(x - 6)

Expanding the equation:

y - 26 = 3x - 18

Simplifying:

y = 3x + 8

So, the equation of the line that passes through (6, 26) with a slope of 3 is y = 3x + 8.

b. To find the equation of a line that passes through two given points (x1, y1) and (x2, y2), we can use the slope-intercept form of the equation:

y = mx + b

where m is the slope and b is the y-intercept. We can find the slope (m) by using the formula:

m = (y2 - y1) / (x2 - x1)

In this case, the points are (5, 5) and (10, 20). Plugging in these values into the formula, we get:

m = (20 - 5) / (10 - 5)
= 15 / 5
= 3

Now, let's find the y-intercept (b) by plugging in one of the points into the slope-intercept form:

5 = 3(5) + b
5 = 15 + b
b = -10

So, the equation of the line that passes through (5, 5) and (10, 20) is y = 3x - 10.

c. Using the same method as above, we can find the equation of the line that passes through (9, 25) with a slope of -3.

m = -3

y = mx + b
25 = -3(9) + b
25 = -27 + b
b = 52

So, the equation of the line that passes through (9, 25) with a slope of -3 is y = -3x + 52.

d. To find the equation of a line that is perpendicular to another line, we need to find the negative reciprocal of the slope of the given line. The given line has the equation:

4x + 6y = 18

To find the slope of this line, we need to rewrite it in slope-intercept form (y = mx + b):

6y = -4x + 18
y = (-4/6)x + 3
y = -(2/3)x + 3

The slope of this line is -(2/3). The negative reciprocal of -(2/3) is 3/2 (flip the fraction and change its sign). This is the slope of the perpendicular line.

Now, we can use the point-slope form of the equation with the given point (1, 1) and the slope 3/2:

y - 1 = (3/2)(x - 1)

Expanding the equation:

y - 1 = (3/2)x - 3/2

Simplifying:

y = (3/2)x - 3/2 + 1
y = (3/2)x - 3/2 + 2/2
y = (3/2)x - 1/2

So, the equation of the line that passes through (1, 1) and is perpendicular to the line 4x + 6y = 18 is y = (3/2)x - 1/2.

e. The equation of a circle can be written in the standard form:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) represents the midpoint of the circle and r represents the radius.

In this case, the midpoint is (2, 2) and the radius is 4. Plugging these values into the standard form, we get:

(x - 2)^2 + (y - 2)^2 = 4^2

Expanding and simplifying:

(x - 2)^2 + (y - 2)^2 = 16

So, the equation of the circle with a radius of 4 and a midpoint at (2, 2) is (x - 2)^2 + (y - 2)^2 = 16.