Problem 76 - Prealgebra

Question

Solve each equation. $$x+x+x=180$$

Step-by-Step Solution

Answer

The solution is \(x = 60\).

Step 1: Simplify the Equation

Combine like terms. The equation is given as \(x + x + x = 180\). This can be simplified by adding the \(x\) terms together, which results in \(3x = 180\).

Step 2: Divide Both Sides

To isolate \(x\), divide both sides of the equation by 3. This gives \(x = \frac{180}{3}\).

Step 3: Calculate the Result

Perform the division \(x = \frac{180}{3}\) to find \(x = 60\).

Key Concepts

Simplifying Expressions

When it comes to algebra, simplifying expressions is a fundamental skill. It involves reducing complex-looking expressions into simpler, more manageable forms. This often means removing unnecessary parts, combining similar terms, or rewriting them in a way that's easier to work with.
For example, in the equation from our exercise, we start with several added terms that can be grouped together: \(x + x + x\). Each \(x\) is a similar term, meaning they can be added together:

• Addition of \(x\) terms: \(x + x + x = 3x\).

By turning multiple terms into a single term, we've simplified the expression, making it easier to solve. Simplifying helps in solving equations faster and with fewer errors. It’s like tidying up a cluttered desk, so you can see clearly what work needs to be done.

Combining Like Terms

In algebra, combining like terms is the process of merging terms in an expression that have the same variable raised to the same power. This is crucial because it reduces complexity and helps us solve equations more easily.
In the given equation, you see the terms \(x + x + x\). Each of these \(x\) terms is similar because they contain the same variable, \(x\), and they are each raised to the power of 1.

• Like terms are combined by performing addition: \(x + x + x = 3x\).

This step significantly simplifies the solving process by reducing the number of terms you are dealing with from three individual \(x\) terms to a single \(3x\). Always check for like terms in an equation, as it's a key step towards simplification and solving.

Division in Equations

Once you've simplified and combined like terms in an equation, the next step often involves isolating the variable to find its value. This is where division in equations comes into play. It involves dividing both sides of the equation by a number that will leave the variable alone on one side.
From the example, after simplifying to \(3x = 180\), you need \(x\) by itself. Here’s the process:

• Divide both sides by 3, which is the coefficient of \(x\): \(x = \frac{180}{3}\).
• This step helps isolate \(x\), enabling you to calculate its exact value.

After doing the division, you get \(x = 60\). Division is pivotal in balancing equations and finding the exact value of unknown variables. Always remember to perform the same division on both sides of the equation to maintain balance.