Question

Which of the following equations might have the graph shown? (More than one answer is possible.)

$$\begin{gathered} \left(a\right)2x+3y=6 \\ \left(b\right)-2x+3y=6 \\ \left(c\right)3x-4y=-12 \\ \left(d\right)x-y=1 \\ \left(e\right)x-y=-1 \\ \left(f\right)y=3x-5 \\ \left(g\right)y=2x+3 \\ \left(h\right)y=-3x+3 \end{gathered}$$

Short answer

The equation which describes the given line is

$$\begin{gathered} \left(b\right)-2x+3y=6 \\ \left(c\right)3x-4y=-12 \\ \left(e\right)x-y=-1 \\ \left(g\right)y=2x+3 \end{gathered}$$

Step-by-step solution

Given information

We are given graph and several equations

Explanation for correct options

Consider the option

$$\begin{gathered} b)-2x+3y=6 \\ y=\frac{2}{3}x+2 \end{gathered}$$

and

$$\begin{gathered} c)3x-4y=-12 \\ 4y=3x+12 \\ y=\frac{3}{4}x+3 \end{gathered}$$

also

$$\begin{gathered} e)x-y=-1 \\ y=x+1 \end{gathered}$$

Similarly

$g)y=2x+3$

Each and every equation has positive y-intercept similarly from the graph we can say that the line has positive y-intercept

Consider the equations

$$\begin{gathered} \left(a\right)2x+3y=6 \\ y=-\frac{2}{3}+2 \\ \left(d\right)x-y=1 \\ y=x-1 \\ \left(f\right)y=3x-5 \\ \left(h\right)y=-3x+3 \end{gathered}$$

In the equation a) and h) the slopes are negative where as in option d) and f) the y-intercept is negative hence this does not match the given graph of lines

Conclusion

The equations that can have the given graph as

$$\begin{gathered} \left(b\right)-2x+3y=6 \\ \left(c\right)3x-4y=-12 \\ \left(e\right)x-y=-1 \\ \left(g\right)y=2x+3 \end{gathered}$$