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Chapter 7: Problem 48

Solve each equation by doing the same thing to both sides. $$ 7.5 a-6=5 a+4 $$

Short Answer

Expert verified
a = 4

Step by step solution

01

Combine Like Terms

First, subtract \(5a\) from both sides to get all the variable terms on one side of the equation:\[7.5a - 5a - 6 = 4\]This simplifies to:\[2.5a - 6 = 4\]
02

Isolate the Variable Term

Next, add 6 to both sides to move the constant term to the other side:\[2.5a - 6 + 6 = 4 + 6\]This simplifies to:\[2.5a = 10\]
03

Solve for \(a\)

Finally, divide both sides by 2.5 to isolate \(a\):\[a = \frac{10}{2.5}\]This simplifies to:\[a = 4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
When solving linear equations, a crucial step is combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the equation \(7.5a - 6 = 5a + 4\), both \(7.5a\) and \(5a\) are like terms because they share the variable \(a\).
Combining them simplifies the equation.
In our exercise, subtract \(5a\) from both sides to gather variable terms on one side:\[7.5a - 5a - 6 = 4\] which simplifies to: \[2.5a - 6 = 4\]
This step ensures we deal with fewer terms, making solving easier.
Isolating Variables
Isolating the variable means to get the variable term alone on one side of the equation. After combining like terms, our equation became \[2.5a - 6 = 4\].
We want \(a\) by itself, so we need to remove \(-6\) from the left side.
We achieve this by adding 6 to both sides:
  • \[2.5a - 6 + 6 = 4 + 6\]
which simplifies to:
  • \[2.5a = 10\]
This isolates the variable term \(2.5a\), preparing it for the final step of solving for \(a\).
Balancing Equations
Balancing equations involves performing the same operation on both sides of the equation. It鈥檚 like being fair鈥攚hatever you do to one side, you must do to the other.
In our example, to isolate the variable, we had to:
  • Subtract \(5a\) from both sides:
    • \[7.5a - 5a - 6 = 4\]
  • Add 6 to both sides:
    • \[2.5a - 6 + 6 = 4 + 6\]
Maintaining balance ensures the equality holds true throughout the process. This concept is fundamental in algebra, keeping both sides equal makes finding the correct solution possible.
Algebraic Manipulation
Algebraic manipulation refers to rearranging and simplifying equations using various algebraic operations. Let鈥檚 break down the final steps:
From \[2.5a = 10\], we need to solve for \(a\).
We do this by dividing both sides by 2.5:
  • \[a = \frac{10}{2.5}\]
This simplifies to:
  • \[a = 4\]
By dividing both sides, we adhere to the balance of the equation while isolating \(a\).
This final manipulation gives us the solution \(a = 4\). Mastering these steps helps in simplifying and solving any linear equation efficiently.

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