Let's solve the given systems of equations using the elimination method.

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Problem 1:

We are given the system:

1. $4x + 3y = 12$
2. $x + 3y = 25$

Step 1: Eliminate $y$

We can eliminate $y$ by aligning the coefficients of $y$ in both equations. The second equation already has $3y$, and the first equation also has $3y$, so we can subtract one equation from the other to eliminate $y$.

We'll subtract equation (2) from equation (1):

$(4x + 3y) - (x + 3y) = 12 - 25$

Simplifying:

$4x + 3y - x - 3y = -13$ $3x = -13$

Solving for $x$:

$x = \frac{-13}{3}$

Step 2: Substitute $x = \frac{-13}{3}$ into one of the original equations.

Substitute $x = \frac{-13}{3}$ into the second equation $x + 3y = 25$:

$\frac{-13}{3} + 3y = 25$

To simplify this, first multiply the entire equation by 3 to eliminate the fraction:

$-13 + 9y = 75$

Solve for $y$:

$9y = 75 + 13$ $9y = 88$ $y = \frac{88}{9}$

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Problem 2:

We are given the system:

1. $x + y = 7$
2. $x - y = 5$

Step 1: Eliminate $y$

We can eliminate $y$ by adding both equations:

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

Simplifying:

$x + y + x - y = 12$ $2x = 12$

Solving for $x$:

$x = \frac{12}{2} = 6$

Step 2: Substitute $x = 6$ into one of the original equations.

Substitute $x = 6$ into the first equation $x + y = 7$:

$6 + y = 7$

Solve for $y$:

$y = 7 - 6 = 1$

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Final Solutions:

1. For the system $4x + 3y = 12$ and $x + 3y = 25$, the solution is: $x = \frac{-13}{3}, \quad y = \frac{88}{9}$

2. For the system $x + y = 7$ and $x - y = 5$, the solution is: $x = 6, \quad y = 1$

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Would you like more details or have any questions about the steps? Here are some related questions for you:

1. How would the elimination method differ if we had different coefficients for $y$ in the two equations?
2. Can the substitution method be used instead of elimination in these systems? How?
3. What happens if one of the equations in a system is inconsistent (e.g., $2x + 3y = 10$ and $2x + 3y = 15$)?
4. How can the elimination method be applied to systems of three equations?
5. What if we had the system with fractional coefficients, how would the process change?

Tip: When using the elimination method, always try to align the coefficients of one variable first. This will make the elimination process smoother and quicker.