Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (x-2 y)(y+3 x)-2 x y+3(x+y-1) $$

Step-by-Step Solution

Answer

The simplified expression is \(3x^2 - 7xy - 2y^2 + 3x + 3y - 3\).

Step 1: Apply distributive property

Multiply the two binomials, (x-2y)(y+3x), using the distributive property (FOIL method): \[ (x-2y)(y+3x) = x(y) + x(3x) - 2y(y) - 2y(3x) \]

Step 2: Simplify the multiplication

Simplify the terms obtained in step 1 and rewrite the expression: \[ = xy + 3x^2 - 2y^2 - 6xy \] Now the overall expression becomes: \[ 3x^2 + xy - 6xy - 2y^2 - 2xy + 3(x+y-1) \]

Step 3: Combine like terms

Add or subtract the coefficients of like terms in the expression: \[ = 3x^2 - 7xy - 2y^2 + 3x + 3y - 3 \]

Step 4: Write the final simplified expression

Arrange the expression in descending degree order (standard form) and write the final simplified expression: \[ = 3x^2 - 7xy - 2y^2 + 3x + 3y - 3 \] The simplified expression is \(3x^2 - 7xy - 2y^2 + 3x + 3y - 3\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions involving multiplication over addition or subtraction. When you see a term like

• \((x-2y)(y+3x)\), it means you need to multiply each part of the first binomial by each part of the second binomial.
• Use the FOIL method, which stands for First, Outer, Inner, Last, to help you remember the steps.

Let's visualize this:

• **First**: Multiply the first terms: \(x \cdot y = xy\)
• **Outer**: Multiply the outer terms: \(x \cdot 3x = 3x^2\)
• **Inner**: Multiply the inner terms: \(-2y \cdot y = -2y^2\)
• **Last**: Multiply the last terms: \(-2y \cdot 3x = -6xy\)

After performing these operations, you'll get a new expression: \( xy + 3x^2 - 2y^2 - 6xy \). It's essential to understand this property to simplify expressions correctly.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Once you've used the distributive property, you'll often end up with several similar terms. These are called "like terms" because they have the same variable raised to the same power.
Here's how to handle like terms:

• Identify terms that have identical variable parts. For instance, in the expression \( xy + 3x^2 - 2y^2 - 6xy - 2xy + 3(x+y-1) \):
  • \(xy\) and \(-6xy\) and \(-2xy\) are like terms because they all have the variables \(x\) and \(y\).
  • Constant terms can also be combined, such as in \(-3+y\).
• Next, add or subtract the coefficients of these like terms:
  • Combine \(xy\), \(-6xy\), and \(-2xy\) to get \(-7xy\).
  • Simplify the expression within any parentheses: \(3(x+y-1)\) becomes \(3x+3y-3\).

By merging these terms properly, the expression simplifies beautifully. This skill helps to ensure that you're not left with an overly complicated answer.

Simplifying Expressions

Simplifying expressions is the final touch in algebra problems, ensuring that your answer is as straightforward as possible. After distributing and combining like terms, you often have a complex expression at first. Let’s break down simplifying:

• First, rewrite the entire expression: \(3x^2 - 7xy - 2y^2 + 3x + 3y - 3\).
• Check that all like terms have already been combined, which confirms we’re on the right track.
• Rewrite the expression in descending order based on the degree of each term. The standard form places terms in order from the highest degree to the lowest:
  • The highest exponent term \(3x^2\) comes first.
  • Next are the terms \(-7xy\) and \(-2y^2\), followed by linear terms \(3x\) and \(3y\), and lastly, the constant \(-3\).

Through these steps, your expression is now neatly simplified and easier to understand: \(3x^2 - 7xy - 2y^2 + 3x + 3y - 3\). Simplification not only makes expressions cleaner but also helps in solving further problems more efficiently.