To solve the system of equations using the elimination method, we want to eliminate one of the variables by adding or subtracting the equations.

Given the system: $4x + y = 7 \quad \text{(Equation 1)}$ $-5x - y = -10 \quad \text{(Equation 2)}$

Step 1: Add the two equations

Notice that the $y$-terms are $+y$ and $-y$, so if we add the two equations together, the $y$-terms will cancel each other out.

$(4x + y) + (-5x - y) = 7 + (-10)$

Simplify both sides: $4x - 5x + y - y = 7 - 10$ $-x = -3$

Step 2: Solve for $x$

$x = 3$

Step 3: Substitute $x = 3$ into one of the original equations

Let's substitute into Equation 1: $4x + y = 7$

Substitute $x = 3$: $4(3) + y = 7$ $12 + y = 7$

Step 4: Solve for $y$

$y = 7 - 12$ $y = -5$

Final Solution:

The solution to the system is $x = 3$ and $y = -5$.

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Would you like more details or have any questions? Here are 5 related problems you could try:

1. Solve the system using elimination: $3x + 2y = 14$, $2x - 2y = 4$.
2. Solve the system using substitution: $2x + y = 5$, $3x - y = 4$.
3. Solve using elimination: $5x + 2y = 20$, $-3x - 2y = -10$.
4. Solve by graphing: $x + y = 4$, $2x - y = 5$.
5. Solve using matrices: $4x + 3y = 10$, $x - y = 2$.

Tip: When using elimination, always aim to cancel one variable by adding or subtracting equations, making it easier to solve for the other variable!