Answer:
Second Option:
[tex]y = -\dfrac{2}{3}x +\dfrac{11}{3}; \:\: y - 1 = \dfrac{3}{2}(x - 4)[/tex]
Step-by-step explanation:
We have two lines on the graph labeled Line A and Line B
Part (a) : Slope intercept form for Line A
The equation of a line in slope-intercept form is
y = mx + b
where
m is the slope - rate of change of y w.r.t x
b is the y-intercept - the value of y at x = 0
To determine slope of a line:
- Take two points lying on the line
- Find the difference in their y-coordinate values (also called rise )
- Find the difference in the corresponding x-coordinate values(also called run)
- Divide rise by run:
Slope = Rise/Run
Mathematically if the two points on a line are labeled (x1, y1) and (x2, y2) then the slope is given by the formula
[tex]m = \dfrac{y2 - y1}{x2-x1}[/tex]
The y-intercept can be computing by taking any point on the line and plugging in (x, y) values and solving for b
Equation For Line A in slope- intercept form
Two convenient points on Line A are (1, 3) and (4, 1)
[tex]m = \dfrac{1 - 3}{4 - 1} = - \dfrac{2}{3}[/tex]
Therefore
Equation of Line A:
[tex]y = -\dfrac{2}{3}x + b[/tex]
Take point (1, 3) and substitute in the above equation:
[tex]3 = -\dfrac{2}{3}\cdot 1 + b\\\\3 = -\dfrac{2}{3}\cdot 1 + b\\\\3 + \dfrac{2}{3} = b\\\\\dfrac{9}{3} + \dfrac{2}{3} = b\\\\\dfrac{11}{3} = b\\\\or\\b = \dfrac{11}{3}[/tex]
Therefore equation Line A is
[tex]y = -\dfrac{2}{3}x +\dfrac{11}{3}[/tex]
We see that this eliminates the first choice which has 2/3 as the slope
Part (b): Equation of Line B in point-slope form
The point slope equation for a line is
y - y1 = m(x - x1)
where
m is the slope
(x1, y1) is any point on the line
We can calculate the slope of Line B as we did for Line A:
Take points(0, -5) and (4, 1) which lie on Line B
[tex]\text{Slope $ m = \dfrac{1 - - 5}{4 - 0} = \dfrac{6}{4} = \dfrac{3}{2}$}[/tex]
So equation of Line B in point slope form
[tex](y - y1) = \dfrac{3}{2}(x - x1)[/tex]
Take point (x1, y1)= (4, 1). This is a convenient point since the coordinates are positive and less confusing to work with
Equation becomes
[tex]y - 1 = \dfrac{3}{2}(x - 4)[/tex]
Therefore the correct choice which corresponds to both equations the second option
[tex]y = -\dfrac{2}{3}x +\dfrac{11}{3}; \:\: y - 1 = \dfrac{3}{2}(x - 4)[/tex]
