Answer: The correct option is graph (A).
Step-by-step explanation: We are given to select the graph that represents the following linear equation :
[tex]y=\dfrac{1}{3}x-4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
Comparing with the slope-intercept form y = mx + c of a straight line, we see that
the slope of the line (i) is [tex]m=\dfrac{1}{3},[/tex] and y-intercept, c = -4.
Also, we know that the slope of a line passing through the points (a, b) and (c, d) is given by
[tex]m=\dfrac{d-b}{c-a}.[/tex]
Graph (A) :
The points (0, -4) and (3, -3) lies on the line. So, its slope will be
[tex]m=\dfrac{-3-(-4)}{3-0}=\dfrac{1}{3},[/tex]
the point where the line crosses the y-axis is (0, -4).
That is, y-intercept, c = -4.
Since the slope and y-intercept of graph (A) is same as that of line (i), so this option is CORRECT.
Graph (B) :
The points (0, -2) and (3, -3) lies on the line. So, its slope will be
[tex]m=\dfrac{-3-(-2)}{3-0}=-\dfrac{1}{3},[/tex]
Since the slope of graph (B) is not same as that of line (i), so this option is NOT CORRECT.
Graph (C) :
The points (0, -4) and (-3, 3) lies on the line. So, its slope will be
[tex]m=\dfrac{3-(-4)}{-3-0}=-\dfrac{7}{3},[/tex]
Since the slope of graph (C) is not same as that of line (i), so this option is NOT CORRECT.
Graph (D) :
The points (-4, 0) and (-3, 3) lies on the line. So, its slope will be
[tex]m=\dfrac{3-0}{-3-(-4)}=\dfrac{3}{1}=3,[/tex]
Since the slope of graph (B) is not same as that of line (i), so this option is NOT CORRECT.
Thus, graph (A) is the correct option.
