Problem 138 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. $$\text { Solve: }-2 x-4=x+5$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=3\)

Step 1: Rewrite the equation

First, rewrite the equation to start solving for x. The equation given is \(-2x-4=x+5\).

Step 2: Move variables to one side

Next, rearrange the equation by moving all variables to one side by subtracting x from both sides. This will produce the equation \(-2x-x=-4-5\).

Step 3: Simplify the equation

After subtraction, simplify the equation which will become \(-3x=-9\).

Step 4: Solve for x

Lastly, solve for x by dividing both sides of the equation by -3. This gives the solution to the equation as \(x=3\)

Key Concepts

Algebraic Manipulation

When solving any linear equation, algebraic manipulation plays a crucial role. This process involves strategically rearranging and simplifying an equation to find the value of the variable.
Let's consider our starting point: the equation \(-2x - 4 = x + 5\). We can see both sides of the equation have terms involving numbers and the variable \(x\).
The goal of algebraic manipulation here is to rearrange these terms to help us isolate \(x\). This means moving terms from one side of the equation to the other, ensuring we perform the same operation on both sides to maintain balance.

• For instance, if you subtract \(x\) from both sides of \(-2x - 4 = x + 5\), then the equation becomes \(-2x - x - 4 = 5\).
• This leaves us with \(-3x - 4 = 5\), thus simplifying the problem at hand by bringing like terms together and making the equation cleaner.

Algebraic manipulation is all about smartly reorganizing the equation to make solving it easier.

Variable Isolation

Variable isolation is the process of getting the unknown variable, often \(x\), on its own on one side of the equation. This is essential in determining its value.
After using algebraic manipulation to reach the form \(-3x - 4 = 5\), we strive to isolate \(x\).
The key is to perform operations that systematically eliminate any other terms connected to \(x\). Here’s a simple way to do this:

• First, eliminate the constant number on the same side as \(x\) by addition or subtraction. For example, add 4 to both sides in our equation, resulting in \(-3x = 9\).
• Next, deal with the coefficient of \(x\). Here it’s multiplying \(x\) by \(-3\), so divide both sides by \(-3\) to isolate \(x\).
• This results in \(x = 3\), meaning the variable \(x\) is now isolated and solved.

Variable isolation is about breaking down the equation step-by-step until the variable stands alone.

Equation Simplification

Simplification in solving equations helps transform a complex equation into a simple one that can be more easily solved.
In our example, we moved from \(-2x - 4 = x + 5\) to \(-3x = 9\) through various steps, each of which involved simplification.
Simplification can take several forms:

• Combining like terms, as we did by subtracting \(x\) from \(-2x\), yielding \(-3x\).
• Eliminating constants or numbers by performing the same arithmetic operation on both sides of the equation, such as adding 4 to obtain \(-3x = 9\).
• Sometimes, it's about cancelling out numbers by division or multiplication to reduce the equation to a more straightforward form. In our final step, dividing by \(-3\) simplified things to \(x = 3\).

The objective of simplification is always to make the path clearer to the variable solution, by reducing the equation to its simplest form.