Problem 43 - Algebra A Combined Function

Question

Simplify the expressions. $$ (-5 a-7 b)+(5 a-8 b) $$

Step-by-Step Solution

Answer

The simplified expression is \(-15b\).

Step 1: Identify Like Terms

In the expression \[ (-5a - 7b) + (5a - 8b) \]you have two sets of terms: terms with \(a\) and terms with \(b\). Group the terms \(-5a\) and \(5a\) together, as they are like terms. Similarly, group \(-7b\) and \(-8b\) together.

Step 2: Combine Like Terms with 'a'

Add the coefficients of the \(a\) terms: \[-5a + 5a = 0a.\] Since both terms cancel each other, their sum is zero.

Step 3: Combine Like Terms with 'b'

Add the coefficients of the \(b\) terms: \[-7b + (-8b) = -15b.\] Remove the parentheses and keep the negative sign, as addition of a negative number is equivalent to subtraction.

Step 4: Write the Simplified Expression

Now, combine both results from the above steps:\[0a - 15b.\]Since \(0a\) represents zero, the final simplified expression can be written as \[-15b.\]

Key Concepts

Like Terms

In algebra, understanding the concept of **like terms** is essential for simplifying expressions. Like terms are terms that have the same variable raised to the same power. Unlike terms, on the other hand, differ in variables or exponents.

For instance, in the expression \((-5a - 7b) + (5a - 8b)\), we identify like terms by looking at the variable parts:

• \(-5a\) and \(5a\) are like terms because they both contain the variable \(a\).
• \(-7b\) and \(-8b\) are like terms as they share the variable \(b\).

Being able to recognize like terms allows you to group them together, which is the first step in simplification. This grouping sets the foundation for the next stage: combining them.

Combining Like Terms

Once like terms are grouped, the next step in simplifying an algebraic expression is **combining like terms**. This involves adding or subtracting the coefficients of these terms. A coefficient is the numerical factor of a term that includes a variable.

For the terms that include \(a\), we calculate:

• \(-5a + 5a\), which simplifies to \(0a\) because \(-5 + 5 = 0\).

When terms cancel each other out this way, it means they have summed to zero, effectively removing that variable from the expression.

For the terms with \(b\), we perform the operation:

• \(-7b + (-8b) = -15b\), by adding \(-7\) and \(-8\), which results in \(-15\).

This step results in a simpler expression that only uses necessary variables, making it easier to interpret further calculations.

Coefficient

A **coefficient** is a number in front of a variable in an algebraic expression that tells you how many times the variable is to be taken. It is integral to the process of combining like terms and simplifying.

In our expression, different terms have coefficients:

• The term \(-5a\) has a coefficient of \(-5\).
• The term \(5a\) carries a coefficient of \(5\).
• For \(-7b\), the coefficient is \(-7\), and for \(-8b\), it is \(-8\).

When simplifying by combining like terms, the coefficients are added or subtracted to reduce the expression. For example,

• In \(-5a + 5a = 0a\), the coefficients \(-5\) and \(5\) add up to zero.
• In \(-7b + (-8b) = -15b\), adding the coefficients \(-7\) and \(-8\) results in \(-15\).

A clear understanding of coefficients allows you to efficiently handle and simplify expressions, leading to accurate results.