Problem 35 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (0.2 x+1.2 y)(0.3 x-2.1 y) $$

Step-by-Step Solution

Answer

The simplified expression is: \[ (0.2x + 1.2y)(0.3x - 2.1y) = 0.06x^2 - 0.06xy - 2.52y^2 \]

Step 1: Write down the given expression

Write down the expression that we need to simplify: \[ (0.2x + 1.2y)(0.3x - 2.1y) \]

Step 2: Apply the distributive property (FOIL Method)

Apply the distributive property (FOIL Method), which stands for First, Outer, Inner, and Last: First: Multiply the first terms of both binomials: \[ (0.2x)(0.3x) \] Outer: Multiply the outer terms of both binomials: \[ (0.2x)(-2.1y) \] Inner: Multiply the inner terms of both binomials: \[ (1.2y)(0.3x) \] Last: Multiply the last terms of both binomials: \[ (1.2y)(-2.1y) \]

Step 3: Multiply the terms

Now, multiply the terms from Step 2: First: \[ (0.2x)(0.3x) = 0.06x^2 \] Outer: \[ (0.2x)(-2.1y) = -0.42xy \] Inner: \[ (1.2y)(0.3x) = 0.36xy \] Last: \[ (1.2y)(-2.1y) = -2.52y^2 \]

Step 4: Combine like terms

Combine like terms obtained in Step 3. In this case, we have both outer and inner terms which are like terms (-0.42xy and 0.36xy): \[ 0.06x^2 - 0.42xy + 0.36xy - 2.52y^2 \] Combine the like terms (-0.42xy and 0.36xy): \[ 0.06x^2 - 0.06xy - 2.52y^2 \]

Step 5: Write the simplified expression

Now the expression is simplified, and we can write it down: \[ (0.2x + 1.2y)(0.3x - 2.1y) = 0.06x^2 - 0.06xy - 2.52y^2 \]

Key Concepts

Distributive Property

The distributive property is a crucial concept in algebra that helps simplify expressions and solve equations. Essentially, it states that multiplying a single term across a set of terms within parentheses spreads that single term over each component within the parentheses. Imagine unpacking your holiday gifts; each term in the grouping gets its fair share! In more formal terms, for any three numbers or variables, say, a, b, and c, the distributive property can be expressed as:\[a(b + c) = ab + ac\]This property allows us to "distribute" a single term outside the parentheses over the terms inside. In the given exercise, the expression \[(0.2x + 1.2y)(0.3x - 2.1y) \]applies the distributive property by managing the multiplication of each pair of terms within the binomials. The key takeaway here: when encountering expressions enclosed in parentheses being multiplied, remember to distribute the outer term to each inner term!

FOIL Method

The FOIL method is often used to apply the distributive property when dealing with binomials, which are algebraic expressions containing two terms. FOIL stands for First, Outer, Inner, Last, and describes the order in which you multiply the terms in each binomial. This organized approach ensures you don't miss any combinations, helping simplify the process considerably. Here’s a breakdown:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of each binomial.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

Applying FOIL to the standard form exercise at hand shows how efficiently each term of one binomial gets "paired" with every term of another binomial, as demonstrated below:First: \( (0.2x)(0.3x) \) yields \( 0.06x^2 \)
Outer: \( (0.2x)(-2.1y) \) yields \( -0.42xy \)
Inner: \( (1.2y)(0.3x) \) yields \( 0.36xy \)
Last: \( (1.2y)(-2.1y) \) yields \( -2.52y^2 \)
Utilizing FOIL helps ensure each pair of terms gets its rightful place in the resulting expression.

Simplifying Expressions

Simplifying expressions is critical in algebra to make calculations more manageable or to work toward solving an equation. It involves combining like terms—terms that have the same variables raised to the same power. In our exercise, after using the FOIL method, we end up with these terms:\[0.06x^2 - 0.42xy + 0.36xy - 2.52y^2\]Here, like terms \(-0.42xy\) and \(0.36xy\) need to be combined. Combining involves summing their coefficients, which simplifies:\[-0.42xy + 0.36xy = -0.06xy\]We then express the entire simplified expression:\[0.06x^2 - 0.06xy - 2.52y^2\]This now is a much simpler form, more reduced than its original multiline statement. Breaking it down step-by-step makes algebra less intimidating and helps tackle more complex expressions with ease! So remember, simplify by collecting like terms, reducing the clutter for a clearer expression that’s easier to work with.