To solve this system of equations graphically, we need to plot the two equations on the provided set of axes and find the intersection point, which will be the solution to the system. Here’s a step-by-step approach to solving it:

Given Equations:

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

Steps to Plot and Solve Graphically:

1. Plot the First Equation: $y = x + 3$

• Slope (m): 1 (meaning for every increase in $x$ by 1, $y$ increases by 1)
• Y-intercept (b): 3 (the point where the line crosses the y-axis)
• Points to Plot:
  • Start at the y-intercept: (0, 3)
  • Using the slope, move right 1 unit in $x$ and up 1 unit in $y$ to get another point: (1, 4)
  • Repeat this to plot multiple points and draw a straight line through them.

2. Plot the Second Equation: $y = -\frac{1}{2}x - 6$

• Slope (m): $-\frac{1}{2}$ (meaning for every increase in $x$ by 2, $y$ decreases by 1)
• Y-intercept (b): -6
• Points to Plot:
  • Start at the y-intercept: (0, -6)
  • Using the slope, move right 2 units in $x$ and down 1 unit in $y$ to get another point: (2, -7)
  • Plot multiple points and draw a line through them.

3. Find the Intersection

• Once both lines are plotted, identify the point where they intersect. This intersection point represents the solution to the system of equations.

Solution:

From graphing these equations, you should see that they intersect at the point $( -6, -3 )$.

So, the solution to the system is: $(x, y) = (-6, -3)$

Would you like more details or have any questions?

Here are some related questions for further practice:

1. How do you determine the slope of a line from an equation?
2. How do you find the y-intercept in a linear equation?
3. How would you solve a system of equations algebraically instead of graphically?
4. How can you check if two lines are parallel by looking at their equations?
5. How do you convert a standard form equation to slope-intercept form?

Tip: Always verify the intersection point by substituting it back into both original equations to ensure it satisfies both.