$a + (a + 2d) = 14$ → $2a + 2d = 14$ → $a + d = 7$ - Coaching Toolbox
Understanding the Equation $ a + (a + 2d) = 14 $: Simplifying to $ a + d = 7 $
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:
$$
a + a + 2d = 14
$$
Image Gallery
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:
🔗 Related Articles You Might Like:
📰 vegas vacation 📰 round trip flights 📰 chase visa login 📰 Windows 10 Boot Into Safe Mode 3208297 📰 This Single Drug At Genoa Pharmacy Is Revolutionizing Health Forever 4599137 📰 Double Ssi Payments December 2025 8151294 📰 Marriott Dallas 5531804 📰 Riverside Convention Center 5793404 📰 Whats Your Taxonomy Code Simple Steps To Find It Fast Easy 4458587 📰 Uncover The Trickiest Online Hide And Go Seek Games Youll Spy Them Before They Hide Again 9122067 📰 You Wont Believe What Happens When You Say Buenas Noches Dios Te Bendiga Every Night 1638271 📰 Automotive Holdings Group Limited Exposes Massive Holdingsheres What Youre Missing 6744991 📰 What Is The Best Weather App 1833697 📰 The Incredibles 3 Shocks Fans What This Final Scene Reveals About The Legends Legacy 6362443 📰 Does Punctuation Go Inside Quotation Marks 7002895 📰 When Change The Time In Usa 1893561 📰 What Cmbel Revealed About His Darkest Mystery Shocked The Worldincredible 8947539 📰 Nj Transit Bus Hours 4083620Final 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.
