Subtract Second from Third: Simplify (9a + 3b + c) − (4a + 2b + c) = 5.2 − 1.5 and Solve for a and b

Understanding how to subtract one algebraic expression from another and simplify complex equations is essential for mastering algebra. In this article, we’ll break down the expression:
(9a + 3b + c) − (4a + 2b + c), simplify it step by step, and solve for the relationship between variables—specifically, how this leads to the equation 5a + b = 0.7.

Understanding the Context

Step 1: Understanding the Expression

We begin with:
(9a + 3b + c) − (4a + 2b + c)

This involves subtracting a second expression from a first one. The second expression is fully enclosed in parentheses and has a negative sign before it, so this represents subtraction.

Step 2: Distribute the Subtraction Sign

Remember: subtracting inside parentheses changes the sign of each term:
(9a + 3b + c) − 4a − 2b − c

Subtract second from third: (9a + 3b + c) − (4a + 2b + c) = 2.5 − 1.8 → 5a + b = 0.7

Key Insights

Now expand fully:
= 9a + 3b + c − 4a − 2b − c

Step 3: Combine Like Terms

Group and simplify terms containing the same variables:

  • For a: 9a − 4a = 5a
  • For b: 3b − 2b = 1b = b
  • For c: c − c = 0

After simplification:
= 5a + b

So:
(9a + 3b + c) − (4a + 2b + c) = 5a + b

Continue Reading

x = 2\cos\left(\frac{\pi}{9}\right),\quad 2\cos\left(\frac{5\pi}{9}\right),\quad 2\cos\left(\frac{7\pi}{9}\right)
But since the question asks for real solutions and the polynomial is cubic, we verify via direct root-finding:
Actually, applying the cubic formula or checking discriminant \( \Delta = -4(-4)^3 - 27(2)^2 = 256 - 108 = 148 > 0 \), which implies **three distinct real roots**.

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Final Thoughts

Step 4: Match with the Right-Hand Side

The original problem states:
(9a + 3b + c) − (4a + 2b + c) = 5.2 − 1.5
But 5.2 − 1.5 simplifies to 3.7, not 5.2 minus 1.8.

Wait — there’s a discrepancy. Let’s double-check the (9a + 3b + c) − (4a + 2b + c) simplification again.

Go back to:
5a + b — this simplifies to 5a + b = ?

Now compare to the problem’s result:
It claims the result equals 0.7, but 5a + b = 0.7 implies:
5a + b = 0.7

So unless additional context or values for a and b are given, 5a + b must equal 0.7, not 3.7.

Where does 0.7 come from?

Possibility: You subtracted numbers incorrectly, or there’s a typo.
The original subtraction yielded:

(9a + 3b + c) − (4a + 2b + c) = 5a + b
But if someone incorrectly computes it and ends up with a number like 3.7, that suggests possibly:
9a + 3b + c − 4a + 2b − c = 5a + 5b = ?
No — signs matter!