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
