Let us solve the given system of equations by graphing:

Given equations:

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

Procedure:

1. Graph the first equation ($y = \frac{1}{2}x + 5$):

   • Slope = $\frac{1}{2}$ (rise/run = 1/2), and y-intercept = $5$.
   • Start at $(0, 5)$ on the graph, then move up 1 unit and right 2 units to find additional points.

2. Graph the second equation ($y = -\frac{5}{2}x - 1$):

   • Slope = $-\frac{5}{2}$ (rise/run = -5/2), and y-intercept = $-1$.
   • Start at $(0, -1)$, then move down 5 units and right 2 units to find additional points.

3. Intersection:

   • The point where these two lines intersect is the solution to the system of equations.

Calculation of Intersection:

Equating the two equations: $\frac{1}{2}x + 5 = -\frac{5}{2}x - 1$

Step 1: Eliminate fractions by multiplying through by 2: $x + 10 = -5x - 2$

Step 2: Combine like terms: $x + 5x = -2 - 10$ $6x = -12$

Step 3: Solve for $x$: $x = -2$

Substitute $x = -2$ into either equation to find $y$. Using the first equation: $y = \frac{1}{2}(-2) + 5 = -1 + 5 = 4$

Solution:

The solution to the system is: $(x, y) = (-2, 4)$

Would you like me to explain graphing further or review anything?

Related Questions:

1. What are the steps for graphing linear equations with fractions?
2. How can you confirm the solution algebraically for any system of linear equations?
3. How do you interpret the slope and y-intercept of a line?
4. How does the graph change when the slope of the line becomes negative?
5. What is the significance of the point of intersection in a real-world context?

Tip:

When solving systems of equations graphically, always verify the intersection point algebraically to ensure accuracy!