Problem 50 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(15 z+1+4 z+2\)

Step-by-Step Solution

Answer

The simplified expression is \(19z + 3\).

Step 1 - Combine Like Terms

Identify the like terms in the expression. In this case, the like terms are the terms with the variable z: 15z and 4z. Combine these terms by adding their coefficients together: \(15z + 4z\).

Step 2 - Add the Numerical Coefficients

Add the coefficients of the like terms: \(15 + 4\). This gives \(19z\).

Step 3 - Combine Constants

Now, combine the constant terms. The constants in this expression are 1 and 2. Add these constants together: \(1 + 2\). This gives 3.

Step 4 - Form the Simplified Expression

Now, put together the terms you've simplified: \(19z\) from the variables and 3 from the constants. The simplified expression is \(19z + 3\).

Key Concepts

Combining Like Terms

One of the essential steps in simplifying algebraic expressions is combining like terms. This means finding and grouping terms that have the same variable raised to the same power.
In the given exercise, the like terms are those with the variable \(z\): 15z and 4z. Notice that both terms contain the variable \(z\).
To combine them, you add their coefficients (the numbers in front of the variables). We get:

• \(15z + 4z = 19z\)

Combining like terms helps reduce the expression to a simpler form, making it easier to understand and work with.

Coefficients

Coefficients are the numbers that are multiplied by the variables in algebraic expressions. In our example, both 15 and 4 are coefficients since they multiply the variable \(z\).
When combining like terms, focus on these coefficients. Add or subtract them to simplify the expression. In the exercise given, the coefficients of the \(z\) terms are 15 and 4. You add them together to get:

• \(15 + 4 = 19\)

Thus, 19 becomes the new coefficient of \(z\), giving us the term \(19z\).
Coefficients play a crucial role in determining the value of terms in algebraic expressions.

Constants

Constants are the numbers in an algebraic expression that do not have a variable attached to them. They stand alone and remain unchanged regardless of the value of the variables.
In our example, the constants in the expression are 1 and 2. These numbers are independent of \(z\) or any other variables.
Combining constants involves simple addition or subtraction. In the exercise, you combine 1 and 2 as follows:

• \(1 + 2 = 3\)

After finding the sum of the constants, you can then include this result in the simplified expression. The final expression, therefore, becomes \(19z + 3\). Constants keep the expression grounded and always have a definitive value.