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Chapter 2: Problem 201

Solve the following equations with variables and constants on both sides. $$21+18 f=19 f+14$$

Short Answer

Expert verified
f = 7

Step by step solution

01

Subtract 18f from both sides

To isolate the variable, subtract 18f from both sides of the equation: 21 + 18f - 18f = 19f + 14 - 18f which simplifies to: 21 = f + 14
02

Subtract 14 from both sides

Next, subtract 14 from both sides to isolate f: 21 - 14 = f + 14 - 14 which simplifies to: 7 = f

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Key Concepts

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

Isolating Variables
One of the primary goals when solving linear equations is to isolate the variable we're solving for. This means getting the variable all by itself on one side of the equation.
In our exercise, we wanted to solve for \(f\) in the equation 21 + 18f = 19f + 14. To do this, we need to perform operations that move all instances of the variable \(f\) to one side of the equation.
First, by subtracting 18f from both sides, we eliminated \(f\) on the left side:
21 + 18f - 18f = 19f + 14 - 18f
This simplifies to:
21 = f + 14.
Now, \(f\) is nearly isolated; we just need to handle the other terms.
Simplifying Equations
Simplifying equations involves combining like terms and performing basic arithmetic operations to make the equation more manageable.
In our problem, simplifying means ensuring both sides of the equation are as reduced as possible, making it easier to isolate the variable.
In the step where we had 21 = f + 14, we recognized that to completely isolate \(f\), we should subtract 14 from both sides:
21 - 14 = f + 14 - 14
Simplifying this, we get:
7 = f
Simplifying at each step not only makes the equation easier to work with but ensures there are fewer chances for errors as we proceed.
Subtraction Property of Equality
The subtraction property of equality is a fundamental rule of algebra which states that if the same amount is subtracted from both sides of an equation, the equality is maintained. This means we can subtract the same value from each side without changing the solution of the equation.
In our given problem, we applied this property twice:
  • First, by subtracting 18f from both sides to start isolating \(f\).
  • Second, by subtracting 14 from both sides to completely isolate \(f\).
This rule is essential as it helps simplify equations and keeps them balanced. Remember, whatever you do to one side, you must do to the other!

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