To solve this system of linear equations, let's break down each equation and see if they can be solved algebraically.

Given: $4x + 7y = 1$ $-x - \frac{7}{4}y = -\frac{1}{4}$

Step 1: Simplify the second equation

Multiply the entire second equation by 4 to eliminate the fraction: $-4x - 7y = -1$

Step 2: Compare the equations

We now have:

1. $4x + 7y = 1$
2. $-4x - 7y = -1$

Add these two equations together: $(4x + 7y) + (-4x - 7y) = 1 + (-1)$ $0 = 0$

This result indicates that the system is dependent, meaning there are infinitely many solutions.

Step 3: Express $y$ in terms of $x$

Since the system is dependent, we can solve one of the equations for $y$ in terms of $x$.

Using the first equation: $4x + 7y = 1$ Solve for $y$: $7y = 1 - 4x$ $y = \frac{1 - 4x}{7}$

Solution

The solution set for this system is: $y = \frac{1 - 4x}{7}$ where $x$ can be any real number.

Would you like me to explain this solution further or verify it graphically?

Here are some related questions you might find helpful:

1. How do you determine if a system of equations is dependent, independent, or inconsistent?
2. What does it mean for two equations to be multiples of each other?
3. Can a system have exactly one solution, and how can you identify such cases?
4. How can substitution be used to solve systems of linear equations?
5. What are the graphical interpretations of dependent and inconsistent systems?

Tip: When solving linear systems, always look for ways to simplify equations, such as clearing fractions, which can make further steps easier.