Let's work through this system of equations to identify the correct graph:

The given system of equations is:

1. $x - 2y = -7$
2. $-4x + y = 14$

We will start by solving both equations for $y$, and then we can identify the graph that corresponds to the solution.

Solving the first equation $x - 2y = -7$:

Rearrange for $y$: $x - 2y = -7 \quad \text{=>} \quad -2y = -x - 7 \quad \text{=>} \quad y = \frac{x + 7}{2}$ So, the first equation in slope-intercept form is: $y = \frac{1}{2}x + \frac{7}{2}$

Solving the second equation $-4x + y = 14$:

Rearrange for $y$: $-4x + y = 14 \quad \text{=>} \quad y = 4x + 14$ So, the second equation in slope-intercept form is: $y = 4x + 14$

Now that we have the equations in slope-intercept form:

1. $y = \frac{1}{2}x + \frac{7}{2}$
2. $y = 4x + 14$

The slopes of the lines are $\frac{1}{2}$ and $4$, respectively, and their y-intercepts are $\frac{7}{2}$ and $14$.

Graph Analysis:

• The first graph shows two lines with distinct slopes, and it seems to intersect at a point. This graph is most likely the correct one, based on the equations.

• The second and third graphs are showing either mismatched slopes or intercepts.

Conclusion:

Based on the slopes and y-intercepts, the third graph most closely represents the system of equations.

Let me know if you need further clarification or have any other questions!

Here are 5 related questions:

1. How do you determine the slope and y-intercept from an equation?
2. What is the meaning of the point where two lines intersect in a system of equations?
3. How do you solve a system of linear equations using substitution or elimination?
4. What is the significance of a system with parallel lines?
5. How do you find the point of intersection for two lines algebraically?

Tip: When graphing a system of equations, always check the slope and y-intercept for each equation to ensure the graph is correct.