To solve this system of equations graphically, we need to plot the two equations and find their point of intersection. Here's the process to solve it:

Given equations:

1. $y = -\frac{5}{6}x - 1$
2. $y = \frac{1}{3}x + 6$

Step 1: Plot the first equation, $y = -\frac{5}{6}x - 1$

• Slope: $-\frac{5}{6}$ means for every increase of 6 units in $x$, $y$ decreases by 5 units.
• Y-intercept: $-1$. This means the line crosses the y-axis at $(0, -1)$.

Step 2: Plot the second equation, $y = \frac{1}{3}x + 6$

• Slope: $\frac{1}{3}$ means for every increase of 3 units in $x$, $y$ increases by 1 unit.
• Y-intercept: $6$. This means the line crosses the y-axis at $(0, 6)$.

Step 3: Find the intersection point

The point where the two lines intersect represents the solution to the system of equations. Let's find this by solving the system algebraically and then confirm it graphically.

1. Set the two equations equal to each other: $-\frac{5}{6}x - 1 = \frac{1}{3}x + 6$
2. Multiply everything by 6 to eliminate the fractions: $-5x - 6 = 2x + 36$
3. Combine like terms: $-7x = 42$
4. Solve for $x$: $x = -6$

Step 4: Substitute $x = -6$ back into one of the original equations to find $y$. Using $y = \frac{1}{3}x + 6$:

$y = \frac{1}{3}(-6) + 6 = -2 + 6 = 4$

Conclusion:

The solution is $x = -6$ and $y = 4$. The lines intersect at the point $(-6, 4)$.

You can plot these two lines and confirm that they intersect at the point $(-6, 4)$.

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Would you like more details or have any questions?

Here are some related questions:

1. How do you graph a line given the slope and y-intercept?
2. What is the significance of the point where two lines intersect?
3. How can you check a graphical solution by solving algebraically?
4. How do you interpret the slope of a line in real-life situations?
5. How can you use graphing tools to solve systems of nonlinear equations?

Tip: When solving systems graphically, always double-check the intersection by plugging values into both equations.