Problem 8 - Prealgebra

Question

Solve the equation. \(-0.2 x-67.3=93.5\)

Step-by-Step Solution

Answer

The solution is \(x = -804\).

Step 1: Isolate the variable term

Start by adding 67.3 to both sides of the equation to remove the constant term from the left side. \(-0.2x - 67.3 + 67.3 = 93.5 + 67.3\), which simplifies to \(-0.2x = 160.8\).

Step 2: Solve for x

Next, divide both sides of the equation by -0.2 to solve for \(x\). \(x = \frac{160.8}{-0.2}\).

Step 3: Calculate the result

Perform the division to find \(x\). \(x = -804\).

Key Concepts

Isolate the Variable

When dealing with equations, an important goal is to isolate the variable. This means getting the variable by itself on one side of the equation. In our example, the equation \(-0.2 x - 67.3 = 93.5\) requires us to focus on moving all numbers off the side where \(x\) is present. To start, you perform actions that simplify the steps later on and make the equation cleaner.

You can do this by adding or subtracting constant terms to both sides of the equation. In this case, since 67.3 is subtracted from the \(-0.2x\), you can add 67.3 on both sides of the equation to maintain equality. By doing so: - The left-hand side becomes \(-0.2x = 160.8\), effectively isolating the term containing \(x\). - This operation gets rid of any extra numbers "hanging around" your \(-0.2x\).Understanding this process is crucial because it helps eliminate distractions around our target—the variable.

Inverse Operations

Inverse operations are essential tools in algebra used for balancing equations. Their purpose is to undo any operations applied to a variable, helping us to isolate that variable effectively. Every mathematical operation has an inverse that reverses it.

- Addition is undone by subtraction and vice versa.- Multiplication can be undone by division, and vice versa.In our equation, once the constant term was removed, we were left with \(-0.2x = 160.8\). Here, the \(-0.2\) is being multiplied by \(x\). To isolate the \(x\), we use the inverse operation of multiplication, which is division. This method will help us remove \(-0.2\) from \(x\) by dividing both sides by \(-0.2\).Embracing inverse operations allows you to simplify complicated expressions to ones we can easily solve.

Division in Equations

Division plays a pivotal role in solving equations, especially when you need to isolate a variable that has a coefficient. Coefficients are numbers multiplied by variables. In the equation \(-0.2x = 160.8\), division enables us to separate \(x\) from its coefficient.

Follow these steps to apply division:- Identify the coefficient of the variable, which is \(-0.2\) in our example.- Divide both sides of the equation by \(-0.2\) to "cancel out" the coefficient, leaving \(x\) by itself. - You are then left with the solution: \(x = \frac{160.8}{-0.2}\).After performing the division, you'll find that \(x = -804\). Be sure to carry out the division carefully, especially with negative numbers.

Understanding division in equations helps you directly solve for the variable, clearing up any confusion about how to proceed once variables and coefficients mix.