$a + (a + 2d) = 14$ → $2a + 2d = 14$ → $a + d = 7$ - Treasure Valley Movers

Understanding the Equation $ a + (a + 2d) = 14 $: Simplifying to $ a + d = 7 $

Solving mathematical equations is a fundamental skill in algebra, and simplifying expressions plays a vital role in making complex problems easier to work with. One such example is the equation:

$$
a + (a + 2d) = 14
$$

This equation appears simple at first glance, but understanding each step of its transformation reveals powerful insights into algebraic manipulation and simplification. Let’s walk through the process step-by-step and explore why simplifying it to $ a + d = 7 $ is both elegant and valuable.

Step 1: Expand the Parentheses

Start by removing the parentheses using the distributive property:

$$
a + a + 2d = 14
$$

Step 2: Combine Like Terms

Combine the like terms $ a + a = 2a $:

$$
2a + 2d = 14
$$

Step 3: Factor Out the Common Term

Notice that both terms have a common factor of 2. Factor it out:

$$
2(a + d) = 14
$$

Step 4: Solve for $ a + d $

Divide both sides by 2:

$$
a + d = 7
$$

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Why Simplifying Matters

Even though the original equation appears more complex, the simplified form, $ a + d = 7 $, reveals a clean linear relationship between variables $ a $ and $ d $. This form is useful because:

• Easier Computation: Working with $ a + d = 7 $ allows faster calculations without extraneous terms.
  
• Greater Flexibility: You can express one variable in terms of the other: $ d = 7 - a $, making substitution simpler in larger equations.
  
• Foundation for Systems: This form is helpful in solving systems of equations, modeling real-world problems, and optimizing under constraints.
  
• Enhanced Readability: Cleaner equations are easier to interpret and validate, reducing errors in complex problem-solving.

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Real-World Application Example