Problem 38 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ x=\frac{-b}{2 a}, \text { for } a $$

Step-by-Step Solution

Answer

The value of \( a \) is \( a = \frac{-b}{2x} \).

Step 1: Understand the Given Formula

The given formula is \( x = \frac{-b}{2a} \) and we need to solve it for the variable \( a \). This means we want to express \( a \) in terms of \( x \) and \( b \).

Step 2: Eliminate the Denominator

To eliminate the denominator, we multiply both sides of the equation by \( 2a \). This gives us \( 2ax = -b \).

Step 3: Solve for 'a'

Now, we want \( a \) alone on one side of the equation. Divide both sides by \( 2x \) to isolate \( a \). This results in \( a = \frac{-b}{2x} \).

Key Concepts

Isolating Variables

When solving equations, one of the key goals is to isolate the variable you are solving for. Isolating a variable means getting it by itself on one side of the equation. This is often necessary when you have an equation with multiple letters. In the context of the problem, we are given the equation \( x = \frac{-b}{2a} \) and need to solve for \( a \). To isolate \( a \), you start by rearranging the equation so that \( a \) is on one side and everything else is on the opposite side. The process involves strategically manipulating the equation, such as through multiplication, division, or even factoring, to achieve the goal.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to solve. This can include a variety of operations such as:

• Adding or subtracting the same value from both sides of the equation
• Multiplying or dividing both sides by the same number
• Using the distributive property to simplify expressions

In our exercise, we used algebraic manipulation when we rearranged the initial equation \( x = \frac{-b}{2a} \) to eliminate \( a \) from the denominator. By multiplying both sides of the equation by \( 2a \), we converted it into an equation without fractions to deal with: \( 2ax = -b \). This simplification is a crucial part of solving equations effectively.

Elimination of Denominators

Whenever you encounter an equation with a fraction, it's often helpful to eliminate the denominator early in the solution process. This simplifies the equation and makes other operations more straightforward. To eliminate the denominator, you multiply each term by the denominator. Through this, you can convert an equation with fractions into one without fractions, simplifying the path to the solution. In the given equation \( x = \frac{-b}{2a} \), multiplying both sides by \( 2a \) clears the fraction, resulting in \( 2ax = -b \). This step simplifies the equation significantly and makes it easier to isolate \( a \) in the final step. By reducing the complexity of the equation, you can solve it more efficiently.