Problem 77 - Elementary Algebra Within Reach

Question

$$ 5 z-5+10 z+2 z+16 $$

Step-by-Step Solution

Answer

The simplified expression is \(17z + 11\).

Step 1: Identify Like Terms

Start by identifying the like terms in the expression. In this case, the like terms are \(5z\), \(10z\), and \(2z\). Likewise, \(-5\) and \(16\) are like terms as they are constants.

Step 2: Combine Like Terms

Sum the coefficients of the variables 'z' which are 5, 10, and 2. And also, sum the constants -5 and 16.

Step 3: Write Down The Final Simplified Expression

Once we have the sums from step 2, write these down to form the final simplified expression.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. It does not have an equality sign like an equation, so it doesn't make a statement of equality. Instead, it is a combination of several components connected through operations such as addition, subtraction, multiplication, and division.

In the original exercise, the expression given was:
\( 5z - 5 + 10z + 2z + 16 \).
This is a typical algebraic expression that consists of terms. Terms are the building blocks of an expression and are separated by plus or minus signs.

• The terms connected to variables here are \(5z\), \(10z\), and \(2z\).
• While constant terms like \(-5\) and \(16\) are standalone numbers without variables.

A crucial skill in working with algebraic expressions is the ability to recognize and combine like terms, which are terms that contain the same variables raised to the same power.

Constants

Constants are specific numbers within an algebraic expression that do not contain any variables. They represent fixed values and remain unchanged regardless of the value assigned to any variables in the expression. Understanding constants is important, especially when simplifying expressions.

In our exercise, the constants are-5 and 16. Even though they don't link to a variable, they still play a role in the expression's value.

• The constant \(-5\) is subtracted from the expression.
• On the other hand, the constant \(16\) is added.

When combining constants, you simply sum or subtract them as you would regular numbers. Here, combining \(-5\) and \(16\) yields:
\(-5 + 16 = 11\).
This understanding helps in creating the final simplified form of an algebraic expression.

Coefficients

Coefficients are the numerical part of terms that include variables. They multiply the variable, giving scale to its contribution within an algebraic expression. For example, in the term \(5z\), the coefficient is 5, which implies the value of the variable \(z\) is multiplied by 5.

In the expression provided by the exercise, identifying and summing coefficients is a fundamental step in combining like terms.

• For \(5z\), the coefficient is 5.
• For \(10z\), the coefficient is 10.
• For \(2z\), the coefficient is 2.

When these terms contain the same variable, we can group their coefficients and sum them up:
\(5 + 10 + 2 = 17\),
giving us a single term \(17z\).
This process helps simplify the expression, making it easier to evaluate or work with further in problem-solving. Knowing how to identify and combine coefficients streamlines the solving of algebraic expressions.