Let's break this problem down step by step.

Part A: Writing an Equation for the First Clue

Diego says: "If we triple the first number and double the second number, the sum is 34."

• Let the first number be $x$.
• Let the second number be $y$.

According to the clue, the equation can be written as:

$3x + 2y = 34$

Part B: Writing an Equation for the Second Clue

Diego then says: "If we take half of the first number and double the second, the sum is 14."

This can be translated into the equation:

$\frac{1}{2}x + 2y = 14$

Part C: Finding the Two Numbers

We now have a system of two equations:

1. $3x + 2y = 34$
2. $\frac{1}{2}x + 2y = 14$

Step 1: Eliminate $y$

Since both equations have the term $2y$, we can eliminate $y$ by subtracting the second equation from the first. First, let's rewrite the second equation to eliminate the fraction by multiplying it by 2:

$x + 4y = 28$

Now, we subtract this equation from the first equation:

$(3x + 2y) - (x + 4y) = 34 - 28$

Simplifying the left side:

$2x - 2y = 6$

This simplifies to:

$x - y = 3$

Step 2: Substitute into One of the Original Equations

We now have $x = y + 3$. Let's substitute this into one of the original equations. Using the equation $3x + 2y = 34$:

$3(y + 3) + 2y = 34$

Simplifying:

$3y + 9 + 2y = 34$

$5y + 9 = 34$

$5y = 25$

$y = 5$

Step 3: Find $x$

Now that we know $y = 5$, substitute it into $x = y + 3$:

$x = 5 + 3 = 8$

Conclusion:

Diego's two numbers are $x = 8$ and $y = 5$.

Explanation:

• When we triple 8 and double 5, we get $3(8) + 2(5) = 24 + 10 = 34$, which satisfies the first equation.
• When we take half of 8 and double 5, we get $\frac{1}{2}(8) + 2(5) = 4 + 10 = 14$, which satisfies the second equation.

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. What other methods could you use to solve the system of equations?
2. What happens if Diego says the sum of the numbers is instead 30? How would that change the system?
3. Can you graph the two equations on a coordinate plane and find their intersection?
4. What are some real-world scenarios where solving systems of equations like this might be useful?
5. How does the elimination method differ from the substitution method?

Tip: Always check your solution by substituting the values back into the original equations to ensure they satisfy all conditions.