Problem 57 - Algebra A Combined Function

Question

\(13 x-5=11 x-11\)

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Identify Like Terms

To solve the equation \(13x - 5 = 11x - 11\), first identify the like terms on both sides. Notice that \(13x\) and \(11x\) are both terms with the variable \(x\), while \(-5\) and \(-11\) are constants.

Step 2: Move Variable Terms to One Side

Subtract \(11x\) from both sides of the equation to get the variable terms on one side: \(13x - 11x - 5 = -11\). This simplifies to \(2x - 5 = -11\).

Step 3: Isolate the Variable Term

Add \(5\) to both sides to isolate the variable term: \(2x - 5 + 5 = -11 + 5\). This simplifies to \(2x = -6\).

Step 4: Solve for the Variable

Divide both sides by \(2\) to solve for \(x\): \(x = \frac{-6}{2}\). This gives \(x = -3\).

Key Concepts

Like Terms

In the world of algebra, 'like terms' are terms that have the exact same variable parts. This makes them comparable and combinable directly. Understanding like terms is crucial when solving equations as it simplifies the process.
When looking at an equation like the one provided:

• "Like terms" refer to the terms in the equation that can be added or subtracted from each other.
• For instance, in the equation \(13x - 5 = 11x - 11\), \(13x\) and \(11x\) are like terms because they both contain the variable \(x\).
• The other terms, \(-5\) and \(-11\), are also considered like terms because they are constants, meaning they do not have a variable attached.

Grasping the concept of like terms helps to consolidate terms efficiently, making it easier to tackle more complex algebraic equations. By identifying and combining like terms, you can reduce the equation to a simpler form, making it much easier to isolate variables and solve equations.

Isolating Variables

Isolating variables involves manipulating an equation to get the variable of interest by itself on one side of the equation. This process is key to identifying the solution in algebraic equations. Let’s take a closer look.
When you're isolating variables:

• Your primary goal is to manipulate the equation so that the variable you are solving for (in our case, \(x\)) stands alone on one side.
• From the step-by-step solution, after identifying like terms, subtracting \(11x\) from both sides gave us the equation \(2x - 5 = -11\). Here, \(x\) is on its way to being isolated.
• Continue to simplify by removing any constant terms on the side of the variable. Adding \(5\) to both sides helps in this example: \(2x = -6\).

This process of isolating the variable paves the way toward finding its value by making the equation manageable. This isolation is vital in understanding how each element in the equation impacts the variable.

Solving Equations

Solving equations is at the heart of algebra, as it involves finding the value of the unknown variable that makes the equation true. The steps involved in solving equations build upon the concepts of like terms and isolating variables. Here's how this process continues.
To solve equations effectively:

• Once you've isolated the variable term like \(2x = -6\), the next step is to solve for the variable by performing the necessary operations to both sides of the equation.
• For this particular equation, dividing both sides by \(2\) results in \(x = \frac{-6}{2}\), which calculates to \(x = -3\).
• This step finalizes the solution as it provides the specific value for \(x\) that satisfies the given equation.

Overall, solving equations involves a sequence of actions that transform the equation step by step until the variable is resolved completely. Mastering this technique allows for tackling various algebraic problems and finding solutions efficiently.