Answer:
1) C. -3/4·x + 3
2) E. 3/4·x - 1
3) I. -2/5·x + 3
4) A. 2/5·x - 1
Step-by-step explanation:
1) The points on the given line are;
(4, -2), (0, 1), and (-4, 4)
The slope of the line = (1 - (-2))/(0 - 4) = -3/4
The y-intercept = (0, 1)
The equation of the line is therefore, y = -3/4·x + 1
The line parallel to the given line has equal slope to the line, and the different y-intercept, therefore, the correct option for the line parallel to the given line is therefore;
C. -3/4·x + 3
2) The equation of the given line is 4·x + 3·y = 12
Rewriting the equation in slope and intercept form gives;
4·x + 3·y = 12
3·y = 12 - 4·x
y = 12/3 - 4/3·x = 4 - 4/3·x
The slope, m₁, of the given line = -4/3
The y-intercept = (0, 4)
A line perpendicular to the given line, has a slope, m₂ = -1/m₁
Where, m₁ = The slope of the given line
The slope of the perpendicular to the given line is therefore;
m₂ = -1/m₁ = -1/(-4/3) = 3/4
Therefore, the equation of the line perpendicular to the given line is of the form
y = 3/4·x + c, where c is a real number
The correct option for the line perpendicular to the given line is therefore; E. 3/4·x - 1
3) The equation of the given line is 2·x + 5·y = 10
Rewriting the equation in slope and intercept form gives;
2·x + 5·y = 10
5·y = 10 - 2·x
y = 10/5 - 2/5·x = 2 - 2/5·x
y = 2 - 2/5·x
The slope, m₂, of a parallel line to the given line is equal to that of the given line, m₁
Whereby from the above equation, we have, m₁ = -2/5
Therefore, m₂ = m₁ = -2/5
The general equation of a line parallel to the given line is therefore;
y = m₂·x + c = -2/5·x + c, where c is a real number
The correct option for the line parallel to the given line is therefore;
I. -2/5·x + 3
4) The points (intercepts) on the given line are (2, 0), and (0, 5)
The slope of the given line is therefore;
(5 - 0)/(0 - 2) = -5/2
The slope of the perpendicular line, m₂ = -1/m₁, gives;
m₂ = -1/(-5/2) = 2/5
The general form equation of the equation of the perpendicular line to the given line is therefore given as follows;
y = m₂·x + c = 2/5·x + c, where c is a real number
The correct option for the line perpendicular to the given line is therefore;
A. 2/5·x - 1.
. The ODE becomes
![\dfrac{\mathrm d}{\mathrm dx}[x^6z]=-6x^4](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5E6z%5D%3D-6x%5E4)
, we get
![x\neq\sqrt[5]{\dfrac{4131}{12}}](https://tex.z-dn.net/?f=x%5Cneq%5Csqrt%5B5%5D%7B%5Cdfrac%7B4131%7D%7B12%7D%7D)
.