Problem 4 - Elementary Algebra

Question

For Problems \(1-10\), find the indicated products by applying the distributive property; for example, $$ \begin{aligned} (x+1)(y+5) &=x(y)+x(5)+1(y)+1(5) \\ &=x y+5 x+y+5 \end{aligned} $$ $$ (x-5)(y+7) $$

Step-by-Step Solution

Answer

\(xy + 7x - 5y - 35\)

Step 1: Apply the Distributive Property

Start by applying the distributive property. This involves distributing each term in the first binomial to each term in the second binomial. For the expression \((x-5)(y+7)\), distribute \(x\) to \(y\) and \(7\), and distribute \(-5\) to \(y\) and \(+7\).

Step 2: Calculate Each Product

Calculate the result of each distributed multiplication:- \(x \cdot y = xy\) - \(x \cdot 7 = 7x\) - \(-5 \cdot y = -5y\) - \(-5 \cdot 7 = -35\)

Step 3: Combine the Results

Now combine all the products from Step 2 to form a single expression: \[xy + 7x - 5y - 35\]

Step 4: Simplify If Possible

Check if the combined expression can be simplified further: Since there are no like terms to combine, \(xy + 7x - 5y - 35\) is the simplest form.

Key Concepts

Binomials Multiplication

The multiplication of binomials might seem complex at first, but it's simply about applying the distributive property. A binomial is an algebraic expression that contains exactly two terms. When multiplying two binomials, like

• \((x-5)\) and \((y+7)\), the key is to ensure that every term in the first binomial multiplies with every term in the second binomial.

Think of it like cross-multiplying, making sure no term is left out.
This step is crucial as it sets the stage for simplifying the expression later on.
By systematically distributing, you ensure a comprehensive approach to multiplication that covers all possible combinations. Use the term-to-term method: multiply the first term of the first binomial by each term of the second, then do the same with the second term of the first binomial.
This results in four distinct products, which are then combined.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and arithmetic operations. Variables represent unknown values that you solve for, and they can change depending on the problem they're being used in.

• When working with algebraic expressions, especially with binomials, a systematic approach is key.

Algebraic expressions like our example of

• \((x-5)(y+7)\), contain multiple terms that you need to manipulate.

By understanding each part, you'll become more adept at recognizing which operations to perform, such as distributing or combining like terms.
Recognizing the structure of expressions and how terms can interact or "combine" is vital as it forms the basis of algebraic manipulation.
Remember, proper notation and order of operations are your best friends in keeping these expressions clear and manageable.

Simplifying Expressions

Once you've multiplied the terms, the next step is simplifying the resulting expression. Simplification in algebra involves combining like terms and ensuring the expression is as concise as possible.
After distributing and obtaining products, like

• \(xy + 7x - 5y - 35\), you'll want to search for terms that can be combined.

Look for similar variables and coefficients. While not every expression will have like terms, spotting them when they do occur is key for simplification.

• In some cases, like in our example, the expression is already in its simplest form.
• Simplification is not about losing information but rather making the expression easier to interpret and work with.

Understand that simplified expressions are often easier to use in further problem-solving or substitution in equations.
With practice, simplifying expressions becomes intuitive and extremely rewarding.