Understanding How to Subtract $75h + 200$ from Both Sides: A Step-by-Step Guide for Algebra

When solving equations in algebra, a common technique is isolating the variable by moving constants to one side of the equation. One foundational but often overlooked step is subtracting $75h + 200$ from both sides of an equation. This approach simplifies complex expressions and strengthens your understanding of equation balancing.

In this article, we’ll explore what it means to subtract $75h + 200$ from both sides of an equation, why this step matters, and how doing so helps maintain equality while simplifying expressions.

Understanding the Context

What Does It Mean to Subtract $75h + 200$ from Both Sides?

At its core, subtracting $75h + 200$ from both sides of an equation ensures that the equation remains balanced. The principle follows from the addition-preservation rule — whatever you do to one side, you must do to the other.

For example, consider the equation:

Subtract $ 75h $ and $ 200 $ from both sides:

Key Insights

$$
x + 75h + 200 = 5h + 500
$$

If we subtract $75h + 200$ from both sides, we get:

$$
x + 75h + 200 - (75h + 200) = (5h + 500) - (75h + 200)
$$

On the left, the $+75h + 200$ cancels out, leaving just $x$. On the right, we simplify further:

$$
x = 5h + 500 - 75h - 200
$$

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Invert to solve:
R_{\text{total}} = 2 \, \Omega
Dr. Rivera, a science administrator, is reviewing a research grant proposal requesting $1.2 million over 4 years. The budget includes $600,000 for personnel, $300,000 for equipment, and the rest for materials and travel. If equipment costs are expected to increase by 10% next year, how much must be allocated for equipment in year 2 to stay within the total grant?

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Final Thoughts

Now combine like terms:

$$
x = (-70h) + 300
$$

Why Subtract $75h + 200$ from Both Sides?

This transformation serves multiple purposes:

  • Isolates the variable — Helping move all constants to one side simplifies finding the value of $h$ or $x$.
  • Balances the equation — Maintains mathematical integrity by preserving equality.
  • Simplifies further steps — Enables easier combination of like terms, making equations easier to solve.

This method is particularly useful in more complex equations involving multiple variables like $h$ or $x$, where immediate isolation of the variable isn’t straightforward.

Real-World Analogy

Think of an equation like a seesaw: both sides must always balance. If you remove the same weight ($75h + 200$) from each side, the perspective (the equation) remains unchanged while revealing new clarity — perhaps exposing the path to the unknown variable.