Problem 39 - Intermediate Algebra

Question

In the following exercises, simplify the following expressions by combining like terms. $$ 10 a+7+5 a-2+7 a-4 $$

Step-by-Step Solution

Answer

The simplified expression is $$22a + 1$$.

Step 1 - Identify like terms

Look for terms that have the same variable and combine them together. In this case, the like terms involve the variable $$a$$ and the constant terms.

Step 2 - Combine terms with $$a$$

Add all the terms with the variable $$a$$: $$10a + 5a + 7a$$.

Step 3 - Simplify terms with $$a$$

Simplify the terms with the variable $$a$$ by adding them: $$10a + 5a + 7a = 22a$$.

Step 4 - Combine constant terms

Add the constant terms together: $$7 - 2 - 4$$.

Step 5 - Simplify constant terms

Simplify the constant terms by adding or subtracting them: $$7 - 2 - 4 = 1$$.

Step 6 - Combine simplified terms

Combine the simplified terms with the variable and the constants together: $$22a + 1$$.

Key Concepts

like terms

In algebra, the term 'like terms' refers to terms in an expression that have the same variable raised to the same power. These terms can be combined to simplify the expression. For example, in the expression \(10a + 7 + 5a - 2 + 7a - 4\), the like terms are those that contain the variable \(a\). Identifying and combining like terms is an essential skill:

• Look for terms that have the same variable and power. For instance, \(10a\), \(5a\), and \(7a\) all have the variable \(a\).
• Constants, such as the numbers \(7\), \(-2\), and \(-4\), are also considered like terms since they don't involve any variables.

Recognizing and combining like terms allows us to simplify expressions easily, making them much more manageable.

combining constants

Combining constants is another fundamental part of simplifying algebraic expressions. Constants are numbers in the expression that do not change and are not attached to variables. In the example \(10a + 7 + 5a - 2 + 7a - 4\), the constants are \(7\), \(-2\), and \(-4\). Follow these steps to combine constants:

• Add and subtract the constants as you see them appearing in the problem. For example, starting with \(7\), you then subtract \(2\) and \(4\).
• Ensure accuracy by performing each addition or subtraction step-by-step: \(7 - 2 = 5\) and \(5 - 4 = 1\).

So, the result of combining the constants \(7\), \(-2\), and \(-4\) is \(1\). This is a critical step before you can finally simplify the entire expression.

variable terms

Variable terms in an algebraic expression are those that contain variables, like \(a\), \(b\), or \(x\). To simplify an expression, we need to combine these terms by performing basic arithmetic. Here’s a closer look into how this is done:

• Identify all the terms with the same variable. For instance, in \(10a + 7 + 5a - 2 + 7a - 4\), \(10a\), \(5a\), and \(7a\) are the variable terms.
• Add the coefficients (the numbers in front of the variables). In our example, this means adding \(10\), \(5\), and \(7\).

The operation can be written as \(10a + 5a + 7a\), and adding these coefficients gives us \(22a\). Now the variable term is simplified to \(22a\), making the expression much clearer and easier to work with in further calculations.