We are given a system of two linear equations: $y = \frac{1}{2}x - 4$ $y = -\frac{5}{4}x + 3$ We need to find the solution to this system of equations, which means finding the values of $x$ and $y$ that satisfy both equations.

1. Problem Description

We are given a system of two linear equations:

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

$y = -\frac{5}{4}x + 3$

We need to find the solution to this system of equations, which means finding the values of $x$ and $y$ that satisfy both equations.

2. Solution Steps

Since both equations are already solved for $y$, we can set the two expressions for $y$ equal to each other:

$\frac{1}{2}x - 4 = -\frac{5}{4}x + 3$

Now, we solve for $x$. First, we can eliminate the fractions by multiplying both sides of the equation by 4:

$4(\frac{1}{2}x - 4) = 4(-\frac{5}{4}x + 3)$

$2x - 16 = -5x + 12$

Next, we add $5x$ to both sides:

$2x - 16 + 5x = -5x + 12 + 5x$

$7x - 16 = 12$

Now, we add 16 to both sides:

$7x - 16 + 16 = 12 + 16$

$7x = 28$

Finally, we divide both sides by 7:

$\frac{7x}{7} = \frac{28}{7}$

$x = 4$

Now that we have the value of $x$, we can substitute it into either of the original equations to find the value of $y$. Let's use the first equation:

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

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

$y = 2 - 4$

$y = -2$

3. Final Answer

The solution to the system of equations is $x = 4$ and $y = -2$.

The solution can be expressed as the ordered pair $(4, -2)$.