Problem 36 - Prealgebra

Question

Solve each equation. Check your solution. $$3 x-5 x=22$$

Step-by-Step Solution

Answer

The solution is \(x = -11\).

Step 1: Combine Like Terms

The given equation is \(3x - 5x = 22\). Start by combining the like terms on the left side of the equation: \(3x - 5x\) simplifies to \(-2x\). Now, the equation becomes \(-2x = 22\).

Step 2: Solve for x

To solve for \(x\), divide both sides of the equation \(-2x = 22\) by \(-2\): \(x = \frac{22}{-2}\). Simplifying gives \(x = -11\).

Step 3: Check the Solution

To verify the solution, substitute \(x = -11\) back into the original equation \(3x - 5x = 22\). Substitute to get: \(3(-11) - 5(-11) = -33 + 55 = 22\), which matches the right-hand side of the equation, confirming that \(x = -11\) is correct.

Key Concepts

Combining Like Terms

When we talk about **combining like terms**, we're discussing a method to simplify algebraic expressions. In any equation, you may find similar terms that can be combined together, which usually are those that have the same variable raised to the same power.
For our equation, you have terms like \(3x\) and \(-5x\). Both these terms involve the variable \(x\) and have the same degree. So, you can easily combine them.

• Firstly, look at the coefficients: 3 and -5.
• The combined coefficient becomes \(3 - 5 = -2\).

After this operation, the expression becomes simply \(-2x\). When done accurately, combining like terms makes solving the equation much easier, allowing you to focus on isolating the variable.

Verifying Solutions

After solving the equation, it's crucial to **verify your solution**. Verification ensures the solution is correct and adheres to the original equation's conditions. Here's how you can verify:

• Take the solution obtained, \(x = -11\), and substitute it into the original equation \(3x - 5x = 22\).
• Replace \(x\) with \(-11\), leading to the expression \(3(-11) - 5(-11)\).
• Calculate each term: \(3(-11) = -33\) and \(-5(-11) = 55\).
• Combine to get \(-33 + 55 = 22\).

Compare \(22\) with the right side of the equation; since they match, the solution is verified. Verification is a good practice that helps in catching any errors made during the solving process.

Negative Numbers

Working with **negative numbers** is a key skill in algebra that can sometimes be challenging for students. When handling negatives:

• Remember that subtracting a number is the same as adding its opposite. For example, \(3x - 5x\) can be thought of as \(3x + (-5x)\).
• When combining, maintain attention to the signs: a positive added to a negative could decrease the value or make it negative, as in the step where \(3 - 5 = -2\).
• In division, like when finding \(x = \frac{22}{-2}\), recall that dividing a positive by a negative gives a negative result. Thus, \(x = -11\).

Understanding these rules will make solving equations with negative numbers much less intimidating. Keeping a firm grip on these concepts is essential as they frequently appear in algebra.