Understanding the Equation $$ 4a = 36 $$ – Solve for a

Mathematics is a foundational skill in problem-solving, and equations like $$ 4a = 36 $$ are common tools used in textbooks, classrooms, and everyday life. In this SEO-optimized article, we’ll break down how to solve the equation $ 4a = 36 $, explain its real-world relevance, and highlight key concepts for students and lifelong learners.

Understanding the Context

What Does $$ 4a = 36 $$ Mean?

The equation $$ 4a = 36 $$ reads as “four times a number a equals thirty-six.” Here, a represents an unknown value that we must determine through algebraic manipulation. Solving for a helps unlock relationships between numbers and strengthens logical reasoning.

Step-by-Step Solution

$$ 4a = 36 $$

Key Insights

To find the value of a, follow these simple algebraic steps:

  1. Start with the equation:
    $$ 4a = 36 $$

  2. Isolate a by dividing both sides by 4:
    $$ a = rac{36}{4} $$

  3. Calculate:
    $$ a = 9 $$

Answer: $$ a = 9 $$

Continue Reading

Solution: Use the Cauchy-Schwarz inequality: $(2^2 + 3^2 + 4^2)(x^2 + y^2 + z^2) \geq (2x + 3y + 4z)^2$. This gives $29(x^2 + y^2 + z^2) \geq 144$, so $x^2 + y^2 + z^2 \geq rac{144}{29}$. Equality holds when $ rac{x}{2} = rac{y}{3} = rac{z}{4} = k$, leading to $x = 2k$, $y = 3k$, $z = 4k$. Substituting into $2x + 3y + 4z = 12$ gives $4k + 9k + 16k = 29k = 12$, so $k = rac{12}{29}$. Thus, the minimum value is $oxed{\dfrac{144}{29}}$.
Question: Find the center of the hyperbola $9x^2 - 18x - 16y^2 + 64y = 144$.
Solution: Complete the square for $x$ and $y$. For $x$: $9(x^2 - 2x) = 9[(x - 1)^2 - 1] = 9(x - 1)^2 - 9$. For $y$: $-16(y^2 - 4y) = -16[(y - 2)^2 - 4] = -16(y - 2)^2 + 64$. Substitute back: $9(x - 1)^2 - 9 - 16(y - 2)^2 + 64 = 144$. Simplify: $9(x - 1)^2 - 16(y - 2)^2 = 89$. The center is at $(1, 2)$. Thus, the center is $oxed{(1, 2)}$.

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Final Thoughts

This means when you multiply 4 by 9, the result is 36—proving the equation holds true.

Why This Equation Matters

Understanding how to solve $ 4a = 36 $ is more than a math exercise—it’s essential in multiple real-world scenarios:

  • Budgeting & Finance: If a monthly subscription costs $4 and your total budget allows $36 for the year, how many months can you afford it? Solving for a tells you the answer is 9 months.
  • Science & Engineering: Many formulas use proportional relationships. For example, calculating speed, density, or scaling in design often relies on equations like $$ 4a = 36 $$.
    -Educational Learning: Mastering linear equations builds problem-solving skills vital in STEM fields.

Tips for Solving Linear Equations

  1. Isolate the variable term – Keep the expression with a alone on one side.
  2. Use inverse operations – Undo multiplication by dividing, or addition by subtracting.
  3. Check your answer – Plug a = 9 back into the original equation: $$ 4(9) = 36 $$, which confirms correctness.

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