Problem 14 - Algebra Form and Function

Question

For each of the linear expressions in \(x\) in exercises. give the constant term and the coefficient of \(x\). $$ 5-2(x+4)+6(2 x+1) $$

Step-by-Step Solution

Answer

Answer: The constant term is 3, and the coefficient of x is 10.

Step 1: Expand the expression

First, we need to expand the expression by multiplying out the brackets and combining like terms: $$ 5 - 2(x + 4) + 6(2x + 1) = 5 - 2x - 8 + 12x + 6 $$

Step 2: Combine like terms

Now, we want to combine the like terms (constants and x terms): $$ -2x + 12x + 5 - 8 + 6 = 10x + 3 $$

Step 3: Identify the constant term and coefficient of x

In our simplified expression, we can easily see the constant term and the coefficient of x: $$ 10x + 3 $$ The constant term is 3, and the coefficient of x is 10.

Key Concepts

Expanding Expressions

When you encounter an algebraic expression with parentheses, such as \( 5 - 2(x+4) + 6(2x+1) \), the first step is often to expand the expression. Expanding means removing the parentheses by distributing any factors outside the parentheses to each term inside. This process helps simplify the expression and makes it easier to work with.

Here's how it works:

• Look at \(-2(x+4)\). Distribute the \(-2\) to both \(x\) and \(4\). This gives you \(-2x - 8\).
• Next, look at \(6(2x+1)\). Distribute \(6\) to both \(2x\) and \(1\). This yields \(12x + 6\).

Now combine these distributed terms with any other terms that were not in parentheses, which gives us:
\(5 - 2x - 8 + 12x + 6\).
With no more parentheses, the expression is fully expanded. This sets the stage for combining like terms.

Combining Like Terms

Combining like terms is essential for simplifying expressions, especially after expanding them. Like terms are terms that have the same variables raised to the same power.

For instance, in the expression \(-2x + 12x + 5 - 8 + 6\), the terms with \(x\) in them are \(-2x\) and \(12x\), and the constant terms are \(5\), \(-8\), and \(6\).

To combine like terms:

• Take the \(x\) terms: \(-2x\) and \(12x\). Add these together: \(-2x + 12x = 10x\).
• Now, look at the constant terms: \(5 - 8 + 6\). Simplify them by adding: \(5 - 8 = -3\) and \(-3 + 6 = 3\).

When combined, these give us the simplified expression:
\(10x + 3\).
The process of combining like terms reduces the number of terms, leading to a simpler and more manageable expression.

Coefficients in Algebra

Understanding coefficients is a crucial part of algebra. Coefficients are numbers that multiply variables in an expression. Consider the simplified expression we derived: \(10x + 3\).

• The number \(10\) is the coefficient of \(x\). It tells us how many times \(x\) is being multiplied.
• The number \(3\), although not a coefficient, is referred to as the constant term because it stands alone without any variables.

Knowing how to identify coefficients helps in understanding the structure of linear expressions and in solving algebraic equations.
In practice, finding the coefficient is simple: look for the number just before the variable in any given term. This understanding is vital for handling more complex algebraic tasks.