To graph the given system of equations, let's first convert each equation into slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Step 1: Rewrite the equations

1. Equation 1: $x - 4y = -1$

   • Solve for $y$: $-4y = -x - 1$ $y = \frac{1}{4}x + \frac{1}{4}$
   • This equation has a slope $m = \frac{1}{4}$ and a y-intercept $b = \frac{1}{4}$.

2. Equation 2: $2x - y = 2$

   • Solve for $y$: $-y = -2x + 2$ $y = 2x - 2$
   • This equation has a slope $m = 2$ and a y-intercept $b = -2$.

Step 2: Plot each line on the graph

1. For the first equation $y = \frac{1}{4}x + \frac{1}{4}$:

   • Start by plotting the y-intercept at $(0, \frac{1}{4})$.
   • From this point, use the slope $\frac{1}{4}$ to find another point by rising 1 unit up and running 4 units to the right.

2. For the second equation $y = 2x - 2$:

   • Start by plotting the y-intercept at $(0, -2)$.
   • From this point, use the slope $2$ to find another point by rising 2 units up and running 1 unit to the right.

Step 3: Find the Intersection Point

After plotting these lines, look for their intersection point. This point will be the solution to the system of equations, where both equations are satisfied.

Would you like more details on any of these steps, or have any questions?

Here are 5 related questions to deepen understanding:

1. How do you find the slope and y-intercept from the standard form of a linear equation?
2. What does the intersection point of two lines represent in a system of linear equations?
3. How can you verify if a point lies on both lines without graphing?
4. How would you graph a line if it’s given in point-slope form?
5. How can you tell if a system of equations has no solution or infinitely many solutions?

Tip: When converting from standard form to slope-intercept form, isolate $y$ carefully, paying attention to the signs!