1.Write an equation of the line containing the given points and parallel to the given line (3,8); x+5y=3
2. Write an equation of the line containing the given points and parallel to the given line (-3,6); 5x=9y+2
3. Write an equation of the line containing the given points and perpendicular to the given line (4,5); 9x+y=7
4. Write an equation of the line containing the given points and perpendicular to the given line (5,-2); 9x+2y=7
5. Media Services charges $30 for a phone and $20/month for its economy plan. Find a model that determines the total cost, C(t), of operating a Media Services phone for t months. C(t)=?
1. (3,8), x + 5y = 3.
m = m2 = -A/B = -1/5.
y = mx + b,
y = (-1/5)3 + b = 8,
b = 8 3/5.
y = (-1/5)x + 8 3/5.
2. Same procedure as #1.
3. (4,5), 9x + y = 7.
m = -9/1 = -9.
m2 = 1/9.
y = (1/9)4 + b = 5,
b = 5 - 4/9 = 4 5/9.
y = (1/9)x + 4 5/9.
4. Same procedure as #3.
5. C(t) = 20t + 30.
1. To find an equation of the line parallel to the given line x + 5y = 3, we need to determine its slope. The given line can be written in slope-intercept form as y = (-1/5)x + 3/5. Since parallel lines have the same slope, the slope of the line we are looking for is also -1/5.
Using the point-slope form of a linear equation, we can write the equation of the line as follows:
y - y₁ = m(x - x₁), where (x₁, y₁) is the given point (3, 8) and m is the slope.
Plugging in the values, the equation becomes:
y - 8 = (-1/5)(x - 3).
Simplifying the equation, we get:
y - 8 = (-1/5)x + 3/5.
2. To find an equation of the line parallel to the given line 5x = 9y + 2, we need to determine its slope. Rearranging the equation to slope-intercept form, we have y = (5/9)x - 2/9. Since parallel lines have the same slope, the slope of the line we are looking for is also 5/9.
Using the point-slope form of a linear equation, we can write the equation of the line as follows:
y - y₁ = m(x - x₁), where (x₁, y₁) is the given point (-3, 6) and m is the slope.
Plugging in the values, the equation becomes:
y - 6 = (5/9)(x - (-3)).
Simplifying the equation, we get:
y - 6 = (5/9)x + 5/3.
3. To find an equation of the line perpendicular to the given line 9x + y = 7, we need to determine its slope. Rearranging the equation to slope-intercept form, we have y = -9x + 7.
Perpendicular lines have slopes that are negative reciprocals of each other. So, the slope of the line we are looking for is 1/9.
Using the point-slope form of a linear equation, we can write the equation of the line as follows:
y - y₁ = m(x - x₁), where (x₁, y₁) is the given point (4, 5) and m is the slope.
Plugging in the values, the equation becomes:
y - 5 = (1/9)(x - 4).
Simplifying the equation, we get:
y - 5 = (1/9)x - 4/9.
4. To find an equation of the line perpendicular to the given line 9x + 2y = 7, we need to determine its slope. Rearranging the equation to slope-intercept form, we have y = (-9/2)x + 7/2.
Perpendicular lines have slopes that are negative reciprocals of each other. So, the slope of the line we are looking for is 2/9.
Using the point-slope form of a linear equation, we can write the equation of the line as follows:
y - y₁ = m(x - x₁), where (x₁, y₁) is the given point (5, -2) and m is the slope.
Plugging in the values, the equation becomes:
y - (-2) = (2/9)(x - 5).
Simplifying the equation, we get:
y + 2 = (2/9)x - 10/9.
5. The total cost, C(t), of operating a Media Services phone for t months can be calculated using the equation:
C(t) = 30 + 20t.
In this equation, 30 represents the cost of the phone and 20t represents the total cost of the economy plan for t months, with a monthly cost of $20. By adding these two costs together, we can determine the total cost of operating a Media Services phone for t months.
1. To find an equation of a line parallel to a given line, we need to determine the slope of the given line. The given line can be rewritten in slope-intercept form (y = mx + b) by isolating y:
x + 5y = 3
5y = -x + 3
y = (-1/5)x + 3/5
Since parallel lines have the same slope, the parallel line will also have a slope of (-1/5). We can use the point-slope form of a line (y - y1 = m(x - x1)) to find the equation of the line containing the given point (3,8):
y - 8 = (-1/5)(x - 3)
Simplifying:
y - 8 = (-1/5)x + 3/5
y = (-1/5)x + 43/5
Therefore, the equation of the line containing the point (3,8) and parallel to the line x + 5y = 3 is y = (-1/5)x + 43/5.
2. Similarly, to find the equation of a line parallel to the given line, we need to determine the slope of the given line. Rearrange the given line equation (5x = 9y + 2) in slope-intercept form:
9y = 5x - 2
y = (5/9)x - 2/9
The slope of the given line is 5/9. Parallel lines have the same slope, so the parallel line will also have a slope of 5/9. Using the point (-3,6), we can write the equation of the line in point-slope form:
y - 6 = (5/9)(x + 3)
Simplifying:
y - 6 = (5/9)x + 15/9
y = (5/9)x + 41/9
Hence, the equation of the line containing the point (-3,6) and parallel to the line 5x = 9y + 2 is y = (5/9)x + 41/9.
3. To find the equation of a line perpendicular to the given line, we need to determine the slope of the given line. Convert the given line (9x + y = 7) to slope-intercept form:
y = -9x + 7
The slope of the given line is -9. Perpendicular lines have slopes that are negative reciprocals of each other. Thus, the perpendicular line will have a slope of 1/9. Using the point (4,5), we can form the equation in the point-slope form:
y - 5 = (1/9)(x - 4)
Simplifying:
y - 5 = (1/9)x - 4/9
y = (1/9)x + 41/9
Therefore, the equation of the line containing the point (4,5) and perpendicular to the line 9x + y = 7 is y = (1/9)x + 41/9.
4. Similar to the previous question, rearrange the given line (9x + 2y = 7) in slope-intercept form:
2y = -9x + 7
y = (-9/2)x + 7/2
The slope of the given line is -9/2. The perpendicular line will have a slope equal to the negative reciprocal of -9/2, which is 2/9. Using the point (5,-2), we can form the equation in point-slope form:
y - (-2) = (2/9)(x - 5)
Simplifying:
y + 2 = (2/9)x - 10/9
y = (2/9)x - 28/9
Hence, the equation of the line containing the point (5,-2) and perpendicular to the line 9x + 2y = 7 is y = (2/9)x - 28/9.
5. The equation to determine the total cost, C(t), of operating a Media Services phone for t months is given by:
C(t) = 30 + 20t
The fixed cost of $30 for the phone doesn't change over time, while the monthly cost of $20 is multiplied by the number of months (t) the phone is in operation. Adding the fixed and variable costs gives the total cost of operating the phone.