Problem 55 - Algebra 1

Question

Simplify the variable expression. $$3 x+2 y-(5 x+2 y)$$

Step-by-Step Solution

Answer

The simplified version of the given expression is \(-2x\).

Step 1: Distribute the Subtraction

Take account of the negative sign in front of the bracket. This means that the sign of each term in the bracket will change when distributed: \(3x+2y-(5x+2y) = 3x+2y-5x-2y\)

Step 2: Combine Like Terms

Resume the expression by combining like terms, i.e., terms with the same variable. So combine the terms with \(x\) and then with \(y\). The expression becomes: \(3x - 5x + 2y - 2y = -2x + 0\)

Step 3: Final Simplification

Now the expression can be further simplified. Since any number added with zero remains the same, the final simplified version of the given expression is \(-2x\).

Key Concepts

Simplification

Simplification in algebra is the process of reducing expressions into their simplest form. We often do this to make calculations easier or to reveal the underlying structure of an expression. Simplification involves several different techniques, including removing brackets, reducing fractions, and combining like terms.

In our example expression, we start with:

1. The given algebraic expression is: \(3x + 2y - (5x + 2y)\).
2. We first deal with the subtraction across a bracket, simplifying it step-by-step by distributing the negative sign.

The final aim is to transform the expression into a form that is easy to understand and work with, which in this case was reduced to just \(-2x\) after simplifying. This simplification process is crucial in algebra to make working with variables and equations more manageable.

Like Terms

"Like terms" are terms in an algebraic expression that have the same variable raised to the same power. This is important because only like terms can be added or subtracted from one another during the simplification process.

For example, in the expression \(3x - 5x + 2y - 2y\):

• \(3x\) and \(-5x\) are like terms because they both contain the variable \(x\).
• \(2y\) and \(-2y\) are also like terms because they contain the same variable \(y\).

To simplify, you add or subtract the coefficients (numbers in front of the letters) of the like terms:
\(3x - 5x \) results in \(-2x\)
\(2y - 2y\) results in 0.

Combining like terms helps in reducing the complexity of the expression, making it more straightforward to solve or interpret.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by terms inside a bracket. This property is especially useful when simplifying complex expressions.

In our problem, we had the expression \(3x + 2y - (5x + 2y)\). The subtraction of \((5x + 2y)\) needs careful attention:

• Distribute the negative sign to both terms inside the bracket.
• \(-1 \times 5x\) becomes \(-5x\)
• \(-1 \times 2y\) becomes \(-2y\)

After applying the distributive property, the expression becomes \(3x + 2y - 5x - 2y\). This step simplifies the expression and makes it easier to combine like terms next. The distributive property, by allowing the distribution of multiplication across terms in a bracket, is key in transforming the original problem into a form where similar terms can be easily identified and combined.