Problem 45 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. Solve \(2.8 a b=5.6 d\) for \(b\).

Step-by-Step Solution

Answer

Answer: \(b = 2\frac{d}{a}\)

Step 1: Understand the equation

The given equation is \(2.8ab = 5.6d\). Our goal is to isolate \(b\) to find its value in terms of \(a\) and \(d\).

Step 2: Divide both sides by 2.8a

To isolate \(b\), we'll divide both sides of the equation by \(2.8a\). This will give us the equation: \(\frac{2.8ab}{2.8a} = \frac{5.6d}{2.8a}\).

Step 3: Simplify both sides of the equation

After dividing both sides by \(2.8a\), we can simplify the equation. The left side simplifies to just \(b\) since the \(2.8a\) terms cancel each other out. The right side simplifies to \(\frac{5.6}{2.8} \times \frac{d}{a}\) as we can divide 5.6 by 2.8. The equation now becomes: \(b = 2\frac{d}{a}\). Now we have found \(b\) in terms of \(a\) and \(d\): \(b = 2\frac{d}{a}\).

Key Concepts

Algebraic Operations

Algebraic operations are the cornerstone of manipulating equations and expressions in mathematics. When we work with an equation like the one provided, we engage in operations such as addition, subtraction, multiplication, and division. These operations allow us to transform and rearrange equations to solve for a particular variable.

In the given problem, solving for variable 'b' requires us to perform a division operation. Division is used to separate 'b' from the other variables and constants it is currently tied to. This operation crucially maintains the equation's balance by equally affecting both sides of the equation. For students, understanding how these algebraic operations affect an equation is essential for mastering the skills needed to manipulate and simplify a variety of algebraic expressions.

Isolating Variables

Isolating a variable, often a primary goal in algebra, means to rewrite an equation so that a single variable stands alone on one side of the equation. To achieve this, it is necessary to apply inverse operations that 'undo' the way a variable is combined with others.

When isolating the variable 'b' from the equation given, we divide both sides of the equation by '2.8a', essentially undoing the multiplication operation that initially involved 'b'. This process demonstrates the significance of understanding the inverse relationships between operations; where multiplication pairs with division and addition with subtraction. By isolating a variable, we are able to clearly see its relationship with other variables, making it easier to solve for 'b' as required in the original problem.

Equation Simplification

Simplification of an equation often involves reducing it to its most basic form by performing operations that do not change the equation's solutions. In the context of the step-by-step solution provided, simplification occurs after dividing both sides by '2.8a'. It's an algebraic critical thinking exercise where we eliminate any unnecessary complexity.

For example, once we've isolated 'b' on one side, simplification helps us see clearly what 'b' equals by minimizing the number of arithmetic steps needed to solve for it. Dividing '5.6' by '2.8' yields '2', and we reorganize the right side of the equation to show a simple ratio of '2d/a'. This simplification narrows our focus just to the relevant variables, and provides a clear and concise representation of how 'b' is related to 'a' and 'd', confirming its importance in solving algebraic equations.