Problem 12 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$ 9 x+5 y-7+4 x-6 y+3(-2) $$

Step-by-Step Solution

Answer

The simplified expression is \(13x - y - 13\).

Step 1: Identify Like Terms

First, let's identify and group the like terms in the expression. We have terms involving \( x \), terms involving \( y \), and constant terms:- Terms with \( x \): \( 9x \) and \( 4x \).- Terms with \( y \): \( 5y \) and \( -6y \).- Constant terms: \( -7 \) and \( 3(-2) \).

Step 2: Distribute Constant Multiplication

Next, distribute the multiplication in the expression \( 3(-2) \) to simplify it to \( -6 \). This gives the expression:\[ 9x + 5y - 7 + 4x - 6y - 6 \]

Step 3: Combine Like Terms for Variables

Now, add together the coefficients of like terms:- For \( x \) terms: \( 9x + 4x = 13x \).- For \( y \) terms: \( 5y - 6y = -y \).

Step 4: Combine Constant Terms

Finally, combine the constant terms:\( -7 - 6 = -13 \).

Step 5: Write the Simplified Expression

Now, put it all together to write the simplified expression:\[ 13x - y - 13 \].

Key Concepts

Like Terms

In algebra, like terms are terms within an expression that have the same variable and can hence be combined. For example, in the expression

• \( 9x + 5y + 4x - 6y \),

- "9x" and "4x" are like terms because both terms contain the variable \(x\). Similarly, "5y" and "-6y" are like terms because they both involve the variable \(y\). Combining like terms is an essential skill in algebra that simplifies expressions and makes them easier to work with. By combining them, you gather all similar terms into a single term. Remember: you can only combine terms if they share the exact pattern of variables.

Distributive Property

The distributive property is a fundamental principle in algebra used to simplify expressions. It allows you to multiply a term across terms within parentheses. For example, in - the expression \( 3(-2) \), you distribute the multiplication, which means multiplying 3 by - (-2), resulting in a simplified product, - \(-6\). This property is handy, especially when dealing with expressions where you need to remove parentheses before combining like terms. In our given expression, utilizing the distributive property to simplify - \(3(-2)\) directly impacts the constant terms we can combine later.

Combining Coefficients

Combining coefficients involves adding or subtracting the numbers in front of the variables to simplify an expression. These numbers are known as coefficients. Consider, for example, the task of combining the coefficients of like terms in the expression\( 9x + 4x \). - Here, you simply add the numerical values, giving you - \( 13x \). Similarly, for the like terms with \( y \), step through \( 5y - 6y \) - and subtract the coefficients, resulting in - \( -y \). The combining of coefficients is essential since it enables you to transform a complicated expression into a simpler one.

Constants in Algebra

Constants in algebra are numbers on their own, without any variables attached. In the expression - \(-7 + (-6)\), these are the numerical values that remain the same. They are vital to remember when simplifying expressions, even though they don’t combine with terms containing variables.To simplify, combine these constant numbers by performing basic arithmetic operations. In our scenario, execute: \(-7 - 6 = -13\). Understanding constants is crucial because they often play a pivotal role in the terms of equations or expressions once variables are isolated.