Problem 12 - Prealgebra

Question

Solve each equation. $$2 x+3.8=-7.7$$

Step-by-Step Solution

Answer

The solution is \( x = -5.75 \).

Step 1: Isolate the Variable

First, we need to get the variable term by itself on one side of the equation. To do this, subtract 3.8 from both sides of the equation: \[ 2x + 3.8 - 3.8 = -7.7 - 3.8 \] This simplifies to: \[ 2x = -11.5 \]

Step 2: Solve for x

Now that we have isolated the term with the variable, divide both sides of the equation by 2 to solve for \( x \): \[ \frac{2x}{2} = \frac{-11.5}{2} \] This gives us: \[ x = -5.75 \]

Key Concepts

Isolating the Variable

To solve a linear equation effectively, our first target is to isolate the variable. This means we want to get the variable by itself on one side of the equation. Imagine you have a basic equation like \(2x + 3.8 = -7.7\). Here, the term with the variable is \(2x\), and we want it alone.

So, what should we do? It's simple! We will perform the opposite operation of what is currently affecting our variable. Since 3.8 is being added to \(2x\), we'll subtract 3.8 from both sides for balance. We perform:

• Subtract 3.8 from both sides: \(2x + 3.8 - 3.8 = -7.7 - 3.8\)

After this step, what remains is \(2x = -11.5\). Now, the variable term is on its own, ready for the next step of solving the equation!

Simplifying Equations

Simplifying equations is like cleaning up a messy room—with each simplification step, the equation becomes clearer and easier to solve. Once you isolate the variable term, you might still need to "tidy up" the equation. Simplification may involve:

• Combining like terms if there are any on either side.
• Reducing fractions when the terms are expressed in fractional form.

In our current example, after isolating \(2x\) we have \(2x = -11.5\). Even though this example doesn’t have any like terms to combine, if it did, we’d combine them now. At this point, the equation is neat enough for the next step—division.

Division of Equations

The final step to solve for the variable, after isolating it, is often division. When the variable is multiplied by a number, as in our case with \(2x\), we use division to get one single \(x\). Think of division as handing out "fair shares".

In the equation \(2x = -11.5\), our goal is to make \(x\) stand alone. To achieve this, divide both sides of the equation by the number multiplying \(x\), which is 2:

• Divide both sides by 2: \(\frac{2x}{2} = \frac{-11.5}{2}\)

This simplifies to \(x = -5.75\). Performing this division gives us the value of \(x\), solving the equation completely. That's how dividing both sides helps us find the solution in linear equations!