Problem 113 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve: \(4=0.25 B\).

Step-by-Step Solution

Answer

B=16

Step 1: Isolate the variable B

In order to isolate the variable B, it would be necessary to divide both sides of the equation by 0.25. This gives \( \frac{4}{0.25} = B \)

Step 2: Simplify the left side of the equation

By performing the division on the left side of the equation, we obtain \( 16 = B \)

Key Concepts

Isolate the Variable

When working with algebraic equations, one of the most fundamental skills to master is learning how to isolate the variable. This involves manipulating the equation so that the variable we're solving for is by itself on one side of the equality. In our exercise, we have the equation 4=0.25 B. The goal here is to find the value of B that makes this equation true.

To do so, we apply the inverse operation to both sides of the equation. Since B is being multiplied by 0.25, we do the opposite action, which is division, to free B from the clutches of 0.25. This is akin to unlocking B from a box to which 0.25 is the lock. By dividing both sides of the equation by 0.25, we are performing the key step in isolating B.

Simple Division in Algebra

Simple division is a fundamental operation in algebra that is often used to solve equations. As with our exercise 4=0.25 B, we see that division is used to isolate the variable and obtain its value. Mathematically, when you divide both sides of an equation by the same non-zero number, you maintain the equation's balance.

In simple terms, imagine you have an evenly balanced seesaw, removing the same weight from both sides keeps it balanced. Algebraically, when we took the step to divide by 0.25, we are removing or 'canceling out' this multiplier to get B alone. After the division, the equation simplifies to B = 16, which is a clear and direct expression of B's value.

Algebraic Equations

Algebraic equations are like puzzles. Each one is a statement saying that two things are equal, and they often contain one or more variables—unknown values we aim to find. The equation from our example, 4=0.25 B, tells us that four times something equals to 0.25 times B. The challenge - and fun - in algebra is finding out what that something is!

In solving these puzzles, we have tools and rules. Such as, we can add, subtract, multiply, and divide both sides of the equation without changing its meaning—so long as we do it equally on both sides. This keeps the equation balanced, like a scale. Think of each step we take as a move that gets us closer to seeing the full picture or solving the puzzle. With each algebra problem we solve, we're not just finding a number; we're training our brains to think logically and systematically.