You have two equations:

1. $y = 3x - 4$
2. $y = \frac{1}{2}x + 3$

To find the solution to this system of equations, we can set the two expressions for $y$ equal to each other because both equations equal $y$:

$3x - 4 = \frac{1}{2}x + 3$

Steps to Solve:

1. Remove the fraction: Multiply through by 2 to eliminate the denominator.

   $2(3x - 4) = 2\left(\frac{1}{2}x + 3\right)$

   Simplifying:

   $6x - 8 = x + 6$

2. Isolate terms involving $x$: Move all $x$-terms to one side and constants to the other.

   Subtract $x$ from both sides:

   $6x - x - 8 = 6$

   Simplifying:

   $5x - 8 = 6$

3. Solve for $x$: Add 8 to both sides:

   $5x = 14$

   Now divide by 5:

   $x = \frac{14}{5} \quad \text{or} \quad x = 2.8$

4. Substitute $x = 2.8$ into one of the original equations: Let's use $y = 3x - 4$.

   $y = 3(2.8) - 4$

   $y = 8.4 - 4$

   $y = 4.4$

Final Solution:

The solution to the system of equations is:

$x = 2.8, \quad y = 4.4$

If you need any further details or have questions, feel free to ask!

Related Questions:

1. How would you solve a system of equations by substitution instead of elimination?
2. What is the graphical interpretation of the solution to a system of linear equations?
3. How can you check if a given solution satisfies both equations in a system?
4. How does the method of elimination work when dealing with systems of linear equations?
5. What is the general formula for solving systems of linear equations using matrices?

Tip:

When solving systems of linear equations, it's helpful to either graph the lines or manipulate the equations algebraically to find where they intersect.