To solve the system of equations:

1. $x + y = 14$
2. $5x - 3y = 38$

we will use the method of combination (also known as the elimination method). Here's how we can proceed:

Step 1: Solve the first equation for one variable

Start with the first equation:

$x + y = 14$

Solve for $x$:

$x = 14 - y$

Step 2: Substitute this expression for $x$ into the second equation

Now, substitute $x = 14 - y$ into the second equation $5x - 3y = 38$:

$5(14 - y) - 3y = 38$

Step 3: Simplify and solve for $y$

Distribute the $5$ on the left side:

$70 - 5y - 3y = 38$

Combine the like terms:

$70 - 8y = 38$

Now, subtract 70 from both sides:

$-8y = 38 - 70$

$-8y = -32$

Finally, divide both sides by $-8$:

$y = \frac{-32}{-8} = 4$

Step 4: Substitute $y = 4$ into the first equation

Now that we know $y = 4$, substitute this back into the first equation $x + y = 14$:

$x + 4 = 14$

Solve for $x$:

$x = 14 - 4 = 10$

Final Answer:

The solution to the system of equations is:

$x = 10, \quad y = 4$

Would you like more details on any of the steps? Or have any questions?

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Related Questions:

1. How can you use substitution instead of elimination to solve this system?
2. What would happen if the system had no solution or infinitely many solutions?
3. How would you solve a system with three variables using elimination?
4. How can you check if your solution is correct by substituting the values back into the original equations?
5. What would the solution look like graphically for this system of linear equations?

Tip:

When using elimination, always aim to eliminate one of the variables by making their coefficients equal and then adding or subtracting the equations.