Solving the Equation: 800(x + 5) + 1200x = 24,000

Mathematics is all around us—hidden in formulas, budgets, and everyday problem-solving. Whether you're a student tackling algebra or a professional figuring out a budget, equations like 800(x + 5) + 1200x = 24,000 come up often. In this detailed SEO-friendly guide, we’ll break down how to solve this linear equation step-by-step, explain its real-world applications, and show why mastering algebra is essential.

Understanding the Context

What Is the Equation: 800(x + 5) + 1200x = 24,000?

At first glance, the equation
800(x + 5) + 1200x = 24,000
might look intimidating. But with careful expansion and simplification, it becomes manageable. This equation involves distributing a compound term, combining like terms, and isolating the variable x to find its exact value.

Step-by-Step Solution: How to Solve 800(x + 5) + 1200x = 24,000

800(x + 5) + 1200x = 24000

Key Insights

Step 1: Expand the Parentheses
Distribute the 800 across (x + 5):
800 × x = 800x
800 × 5 = 4,000
So,
800(x + 5) = 800x + 4,000

Now the equation becomes:
800x + 4,000 + 1200x = 24,000

Step 2: Combine Like Terms
Combine the x terms (800x + 1200x = 2000x):
2000x + 4,000 = 24,000

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Question: If a linguist generates a 3-letter word using distinct letters from the English alphabet, what is the probability that it contains at least one vowel (A, E, I, O, U)?
Solution: Total letters: 26. Vowels: 5, consonants: 21. Total 3-letter arrangements with distinct letters: $26 imes 25 imes 24$. Unfavorable cases (no vowels): $21 imes 20 imes 19$. Favorable probability: $1 - rac{21 imes 20 imes 19}{26 imes 25 imes 24} = 1 - rac{7980}{15600} = 1 - rac{133}{260} = rac{127}{260}$. $oxed{\dfrac{127}{260}}$
Question: If a historian selects 3 manuscripts at random from a collection of 15, where 5 are from the 18th century, what is the probability that all 3 selected are 18th-century manuscripts?

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Final Thoughts

Step 3: Isolate the x-Term
Subtract 4,000 from both sides:
2000x = 24,000 – 4,000
2000x = 20,000

Step 4: Solve for x
Divide both sides by 2000:
x = 20,000 ÷ 2000
x = 10

Verification

Plug x = 10 back into the original equation:
800(10 + 5) + 1200(10) = 24,000
800(15) + 12,000 = 24,000
12,000 + 12,000 = 24,000 ✅

True—our solution works!

Real-World Applications of Linear Equations

Equations like this aren’t just abstract math—they model real-life situations: