First + Fifth = $a + (a + 4d) = 2a + 4d = 14$ - Decision Point

Understanding the Context

What Does the Equation Mean?

At its core, the equation First + Fifth = a + (a + 4d) = 2a + 4d = 14 uses “First” and “Fifth” metaphorically to represent values in an arithmetic sequence—specifically, terms involving a variable a and a constant multiple of a differing constant d. While “First” and “Fifth” don’t appear in standard math textbooks, using symbolic terms helps us model patterns or relationships where change is consistent (like a linear progression).

This expression breaks down as:

• First = a
  
• Fifth = a + 4d
  
• Sum = a + (a + 4d), which simplifies to 2a + 4d
  
• This sum equals 14:

$$
2a + 4d = 14
$$

Key Insights

Why Solve Equations Like This?

Algebraic equations like 2a + 4d = 14 form the backbone of problem-solving across many disciplines. Whether calculating budgets, planning timelines, or modeling scientific data, learning to isolate variables and manipulate expressions is crucial. Understanding these patterns boosts math confidence and develops critical thinking.

Step-by-Step: Solving 2a + 4d = 14

Goal: Find values of a and d that satisfy the equation—though note: since there is one equation with two variables, we typically express one variable in terms of the other.

Final Thoughts

1. Start with the simplified equation:
   $$
   2a + 4d = 14
   $$

2. Simplify further by dividing every term by 2:
   $$
   a + 2d = 7
   $$

3. Solve for one variable:
   For example, isolate a:
   $$
   a = 7 - 2d
   $$

This means a depends on the value of d. For every d, you can calculate the matching a. Try plugging values:

• If d = 1, then a = 5.
  Check: a + (a + 4d) = 5 + (5 + 4×1) = 5 + 9 = 14. ✅
  
• If d = 2, then a = 3.
  Check: 3 + (3 + 8) = 3 + 11 = 14. ✅

Thus, infinitely many solutions exist along the line a + 2d = 7 in the a–d plane.