Problem 31 - Elementary Algebra

Question

Solve. $$ 5.6 a=-39.2 $$

Step-by-Step Solution

Answer

The value of \(a\) is \(-7\).

Step 1: Identify the Problem Type

We are given a simple linear equation, \(5.6a = -39.2\), where we need to solve for the variable \(a\). Our goal is to isolate \(a\) to find its value.

Step 2: Isolate the Variable

To isolate \(a\), divide both sides of the equation by \(5.6\). This gives us:\[a = \frac{-39.2}{5.6}\]

Step 3: Perform the Division

Calculate the division on the right side of the equation:\[a = \frac{-39.2}{5.6} = -7\]Thus, \(a = -7\).

Key Concepts

Solve for a Variable

When solving for a variable in an equation, our goal is to find the value of that variable which makes the equation true. In the original exercise, we had a simple linear equation, which is an equation that involves only the first power of the variable. Let's break down the process.

Consider the equation given: \(5.6a = -39.2\). Here, \(a\) is the variable we need to solve for. Solving for \(a\) means we need to find out what value \(a\) equals that satisfies this equation. To solve for a variable, you want to manipulate the equation until you have the variable alone on one side (commonly, the left side).

Always check your solution by plugging the found value back into the original equation. This ensures your solution is correct.

Isolate the Variable

Isolating the variable is a fundamental step in solving equations. In this step, you want to arrange your equation so that the variable is by itself on one side of the equation.

Consider our example \(5.6a = -39.2\). To isolate \(a\), you need to eliminate the 5.6 that is multiplying it. You can do this by performing the opposite operation: division.

• To "cancel out" the 5.6 multiplying \(a\), divide both sides of the equation by 5.6.
• This helps to keep the equation balanced, meaning both sides change equally, maintaining equality.

After dividing, you get \(a = \frac{-39.2}{5.6}\). Now, \(a\) is isolated, sitting alone on the left side of the equation.

Division in Algebra

Division is a critical mathematical operation in algebra necessary for solving linear equations. When you divide both sides of an equation by the same non-zero number, you're performing a legal algebraic operation to manipulate the equation without altering the equality.

In our original exercise, we performed the division \(a = \frac{-39.2}{5.6}\). Let's fully understand what happens:

• We're dividing \(-39.2\) by \(5.6\) to simplify the equation and isolate \(a\).
• The result of this division is \(-7\), giving us the value for \(a\).
• Always perform division carefully to avoid mistakes, especially with decimals.

Division helps unravel the problem and discover the unknown variable's value.