Question

For the equation -4y =8x, what is the constant of variation?

Answer 1

Answer:

1) Option b is correct, constant of variation is -2.

2) Option c is correct, Equation of line is $y=\frac{2}{3}x+9$

3)Option d is correct, equation of line in a slope-intercept form is, y=2x -5

Step-by-step explanation:

1)

Direct variation states that a relationship between the two variables. we say y varies directly as x

i.e, $y \propto x$ or y =kx.................[1] ;  where k is the constant of variation.

Given the equation: $-4y = 8x$

Divide both sides by -4 we get;

$\frac{-4y}{-4}=\frac{8x}{-4}$

Simplify:

$y = -2x$             .......[2]

On comparing equation [2] with [1] we get;

k = -2

Therefore, the constant of variation is -2

2)

Equation of line is in the form of y =mx +b where m is the slope of the line and b is the y-intercepts.

Given: slope(m)=$\frac{2}{3}$ and y-intercept(b) = 9

Substitute in the equation of line we get;

$y=\frac{2}{3}x+9$

therefore, the equation of line is, $y=\frac{2}{3}x+9$

3)

Slope intercept form: For any two point $(x_1, y_1)$ and $(x_2, y_2)$, the equation of line is given by:

$y -y_1 = \frac{y_2-y_1}{x_2-x_1}(x-x_1)$ or

y= mx + b ; where m is the slope given by

$m =\frac{y_2-y_1}{x_2-x_1}$ and y-intercept $b=(-mx_1 +y_1)$

Given the points (1 , -3) and (3, 1) ;

Then, the equation of line is:

$y-(-3) = \frac{1-(-3)}{3-1}(x-1)$ or

$y+3 = \frac{1+3)}{3-1}(x-1)$

$y+3 = \frac{1+3)}{3-1}(x-1)$

$y+3 = \frac{4)}{2}(x-1)$

Simplify:

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

Using distributive property i.e, $a\cdot(b+c) = a\cdot b + a\cdot c$

y+3 = 2x -2

Subtract 3 from both sides we get;

y+3-3=2x-2-3

Simplify:

y = 2x - 5.

Therefore, the equation of line in a slope-intercept form is y=2x -5

Answer 2

1. -4y = 8x
     y = -8/4x
     y = -2x,,,,constant of variation is -2

2. slope = 2/3, y int = 9........y = 2/3x + 9 <==

3. (1,-3)(3,1)
     slope = (1 - (-3) / (3 - 1) = (1+ 3) / 2 = 4/2 = 2 
  
     y = mx + b
    slope(m) = 2
    use either point (3,1)...x = 3 and y = 1
    now sub and find b, the y int
    1 = 2(3) + b
    1 = 6 + b
    1 - 6 = b
     -5 = b
  
    so ur equation is : y = 2x - 5 <==