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:

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

Understanding the Context

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:

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

$a + (a + 2d) = 14$ → $2a + 2d = 14$ → $a + d = 7$

Key Insights

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:

Continue Reading

Factor out \( t \):
t(t^2 - 5t + 6) = 0
This gives the factors \( t = 0 \) and \( t^2 - 5t + 6 = 0 \). Solve the quadratic equation:

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Final Thoughts

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

Step 4: Solve for $ a + d $

Divide both sides by 2:

$$
a + d = 7
$$

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.

Real-World Application Example