Problem 26 - Elementary Algebra

Question

Perform the following additions and subtractions. $$ (9 m-n)-(10 m+12 n) $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $(9m - n) - (10m + 12n)$ Answer: $-1m -13n$

Step 1: Distribute the negative sign

First, distribute the negative sign outside the second parentheses to each term inside: $$ (9m - n) - (10m + 12n) = 9m - n - 10m - 12n $$

Step 2: Combine like terms

Now, combine the like terms, which are the terms containing m and the terms containing n: $$ (9m - 10m) + (-n - 12n) $$

Step 3: Perform the additions and subtractions

Finally, perform the required additions and subtractions for each set of like terms: $$ (-1m) + (-13n) $$ Thus, the simplified expression is: $$ -1m -13n $$

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. Variables represent unknown values and are usually denoted by letters, such as the \( m \) and \( n \) in our exercise. These expressions can take on different values depending on the values assigned to their variables.

When working with them, it's crucial to understand that terms with the same variables raised to the same power, such as \( 9m \) and \( -10m \) can be combined through addition or subtraction—this is referred to as 'combining like terms'. However, terms with different variables or exponents, like \( m \) and \( n \) in the given exercise, cannot be combined in the same way since they represent different quantities or dimensions.

Understanding how to manipulate these expressions is fundamental in algebra and is a stepping stone for solving equations and inequalities, which are a central part of the subject.

Addition and Subtraction of Polynomials

Polynomials are algebraic expressions that consist of several terms, and each term is a product of a constant and a variable raised to a non-negative integer exponent. In the case of the given exercise, we have a polynomial subtraction: \((9m - n) - (10m + 12n)\).

When adding or subtracting polynomials, one must align like terms—those with the same variables raised to the same power—and add or subtract their coefficients. In our example, during the combining stage, the coefficients of \(m\) terms, \( 9 \) and \(-10\), are subtracted, as are the coefficients of the \(n\) terms, \(-1\) and \(-12\), leading to the final simplified expression.

Importance of Order and Signs

Keep in mind that order matters, especially regarding the subtraction of polynomials. Also, the subtraction of a polynomial is equivalent to adding its additive inverse, meaning each term of the subtracted polynomial must get its sign changed before combining like terms.

Distributive Property

The distributive property is a cornerstone of algebra and offers a way to simplify expressions that involve multiplication combined with addition or subtraction. Using this property, you can 'distribute' a multiplier to each term within parentheses.

In our exercise, the distributive property is applied in the first step when the negative sign from the subtraction outside the parenthesis is distributed to both terms inside, turning \(+(10m+12n)\) into \(-10m-12n)\). This switch is essential for setting up the problem correctly and prevents errors during the combining like terms process.

Understanding the Property

Mathematically, the distributive property is expressed as \( a(b+c) = ab + ac \) . It's important to apply this property correctly to avoid mistakes in algebraic manipulation, especially when dealing with negatives, as seen in subtracting polynomials.