Problem 36 - Algebra 2

Question

Simplify each expression. $$ 7 a+3 b-4 a-5 b $$

Step-by-Step Solution

Answer

The simplified expression is 3a - 2b.

Step 1: Identify like terms

Observe the expression and identify like terms. Like terms are terms that have the exact variable parts. In this expression, there are terms with 'a' and terms with 'b': - Terms with 'a': 7a and -4a - Terms with 'b': 3b and -5b.

Step 2: Combine like terms with 'a'

Combine the terms with the variable 'a'. This involves adding or subtracting the coefficients of these terms. The terms are 7a and -4a, so combine them by performing the operation:\[7a - 4a = (7 - 4)a = 3a\]

Step 3: Combine like terms with 'b'

Next, combine the terms with the variable 'b'. The terms are 3b and -5b. Combine them by performing the operation:\[3b - 5b = (3 - 5)b = -2b\]

Step 4: Write the simplified expression

Combine the results from Steps 2 and 3 to write the simplified expression. You have 3a from the 'a' terms and -2b from the 'b' terms. Thus, the simplified expression is:\[3a - 2b\]

Key Concepts

Like Terms in Algebraic Expressions

In algebra, the concept of like terms is crucial when it comes to simplifying expressions. Like terms are terms that contain the same variables raised to the same power. It means they have identical variable parts but can have different coefficients. For instance, in the expression \(7a + 3b - 4a - 5b\), you can group terms that have the same variable: those with 'a' are like terms and those with 'b' are also like terms. Understanding like terms is the first step in the process of algebraic simplification, allowing you to simplify expressions efficiently by lowering the number of terms you'll need to work with.

Combining Like Terms

Once you've identified the like terms, the next step is to combine them. This process simplifies the expression and makes it more manageable. To combine like terms, you add or subtract their coefficients while keeping the variable part the same.

For example, when combining the like terms from the expression \(7a + 3b - 4a - 5b\):

• Combine \(7a\) and \(-4a\) to get \(3a\)
• Combine \(3b\) and \(-5b\) to get \(-2b\)

You essentially perform the operations on the numbers in front of the variables, which simplifies the expression. By combining like terms, the expression becomes more concise and easier to interpret.

Understanding Coefficients in Expressions

Coefficients are the numerical parts of terms in an algebraic expression. They represent how many times the variable is being multiplied. In the expression \(7a + 3b - 4a - 5b\):

• 7 is the coefficient of \(a\) in \(7a\)
• 3 is the coefficient of \(b\) in \(3b\)
• -4 is the coefficient of \(a\) in \(-4a\)
• -5 is the coefficient of \(b\) in \(-5b\)

When combining like terms, you'll be adding or subtracting these coefficients. In doing so, you exchange complex, lengthy expressions for more compact and understandable forms by performing arithmetic operations on the coefficients.

Efficient Algebraic Simplification Techniques

Algebraic simplification reduces expressions to their simplest forms, making them easier to work with. This process involves several techniques, but identifying and combining like terms is a primary method. The goal of simplification is to transform the expression so that it is as short and simple as possible, while keeping its original value. To effectively simplify expressions, it’s crucial to:

• Look for like terms, those with matching variable parts.
• Perform arithmetic operations (addition or subtraction) on their coefficients.
• Restructure the expression, writing it out with the newly calculated values.

By mastering these techniques, you can handle more complex algebraic expressions with confidence, thus paving the way for more advanced mathematical problem-solving.