The equation of the line we have been asked to plot is:
[tex]y=\frac{2}{3}x+3[/tex]First of all, let us compare this equation to the standard equation of a line. The standard equation is given by:
[tex]\begin{gathered} y=mx+c \\ m=\text{slope} \\ c=y-\text{intercept (the value of y when x = 0)} \end{gathered}[/tex]Hence we can conclude that:
[tex]\begin{gathered} \text{slope(m)}=\frac{2}{3} \\ y-\text{intercept(c)}=3 \end{gathered}[/tex]Whenver the value of the slope is positive as it is in this case, then the graph should move upwards from left to right. i.e. /.
Hence, Option D is wrong.
Also, we have already stated that y-intercept (c) is where the graph crosses the y-axis or when x = 0.
Therefore, since c = 3, we can further eliminate Option B because it crosses the y-axis at -3 instead of 3.
Finally in order to choose what the answer is between Options A and C, we should substitute
y = 0 into the equation to determine the equation when the graph crosses the x-axis (i.e. when y = 0)
This is done below:
[tex]\begin{gathered} y=\frac{2}{3}x+3 \\ \text{substitute y= 0} \\ \\ 0=\frac{2}{3}x+3 \\ \text{subtract 3 from both sides} \\ -3=\frac{2}{3}x \\ \\ \text{ multiply both sides by}\frac{3}{2} \\ \\ -3\times\frac{3}{2}=\frac{2}{3}x\times\frac{3}{2} \\ \\ \therefore x=-\frac{9}{2}=-4.5 \end{gathered}[/tex]This means that the graph passes through the x-axis at -4.5.
The only option that has this characteristic out of Options A and C is Option C.
Therefore, the final answer is Option C
