Q. 130 - Precalculus Enhanced with Graphing Utilities

Question

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

$$\begin{gathered} \left(a\right) 2x + 3y = 6 \\ \left(b\right) 2x - 3y = 6 \\ \left(c\right) 3x + 4y = 12 \\ \left(d\right) x - y = 1 \\ \left(e\right) x - y = -1 \\ \left(f\right) y = -2x - 1 \\ \left(g\right) y = - \frac{1}{2} x + 10 \\ \left(h\right) y = x + 4 \end{gathered}$$

Step-by-Step Solution

Answer

The following equations describes the given line

$$\begin{gathered} \left(a\right) 2x + 3y = 6 \\ \left(c\right) 3x + 4y = 12 \\ \left(g\right) y = -\frac{1}{2} x + 10 \end{gathered}$$

Step 1: Given information

We are given graph and several equations.

Step 2: Step 2: Explanation for correct options

The lines shown in the graph has positive y-intercept and also the slope of the line is negative 

Consider the equation

$$\begin{gathered} \left(a\right) 2x + 3y = 6 \\ y=-\frac{2}{3}+6 \\ \left(c\right) 3x + 4y = 12 \\ y=-\frac{3}{4}x+3 \\ \left(g\right) y = -\frac{1}{2}x + 10 \end{gathered}$$

In all the above options the slope is negative and y-intercept is positive

Hence the graph can describe the given lines

Step 3: Explanation for other options

Consider the equations

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

From the above options options b) ,d) ,e),h) have positive slope

And option f) has negative y-intercept

Hence the graph can not be described by these lines

Step 4: Conclusion

The graph can describe the given lines 

$$\begin{gathered} \left(a\right) 2x + 3y = 6 \\ \left(c\right) 3x + 4y = 12 \\ \left(g\right) y = -\frac{1}{2} x + 10 \end{gathered}$$