We are tasked with solving the given system of linear equations:

$-2x - 5y = -29$ $4x + y = 13$

Step 1: Solve one equation for a variable

From the second equation, solve for $y$: $y = 13 - 4x$

Step 2: Substitute into the first equation

Substitute $y = 13 - 4x$ into the first equation: $-2x - 5(13 - 4x) = -29$ Simplify: $-2x - 65 + 20x = -29$ Combine like terms: $18x - 65 = -29$ Solve for $x$: $18x = 36 \quad \Rightarrow \quad x = 2$

Step 3: Solve for $y$

Substitute $x = 2$ into $y = 13 - 4x$: $y = 13 - 4(2) = 13 - 8 = 5$

Final Answer:

The solution to the system is: $(x, y) = (2, 5)$

Would you like further details or explanations about these steps? Here are five related questions for practice:

1. Can you verify the solution by substituting $x = 2$ and $y = 5$ back into both equations?
2. What is the graphical interpretation of solving a system of equations?
3. How would the solution change if the equations were inconsistent?
4. Could you use the elimination method to solve this system instead?
5. How can you confirm if a system has infinite solutions?

Tip:

Always simplify one equation to express one variable in terms of the other—it often makes substitution easier!