Problem 43 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 1 x+5+4 x-22 x-2-1 x+5 $$

Step-by-Step Solution

Answer

The simplified expression is \(-18x + 8\).

Step 1: Identify Like Terms

To simplify the expression, look for terms that have the same variable. In our expression, these terms are the ones involving x: \(1x, 4x, -22x, -1x\). All other terms are constants without variables and can be grouped together.

Step 2: Combine the 'x' Terms

Add or subtract the coefficients of the like terms involving 'x'. Start by listing them: \(1x, 4x, -22x, -1x\). Simplify: \(1 + 4 - 22 - 1\), which equals \(-18\). This gives us \(-18x\).

Step 3: Simplify Constant Terms

Next, deal with the constant terms: \(5 - 2 + 5\). Simplify these: \(5 - 2 + 5 = 8\).

Step 4: Form the Final Expression

Combine the result from Step 2 and Step 3 to form the final simplified expression. The simplified expression is \(-18x + 8\).

Key Concepts

Like Terms

In algebra, identifying like terms is crucial to simplifying expressions effectively. Like terms are those terms in an expression that have the same variables raised to the same powers. For example, in the expression

• \(1x + 5 + 4x - 22x - 2 - 1x + 5\)

the terms involving 'x' such as \(1x, 4x, -22x, -1x\) are considered like terms.
Even though these terms have different coefficients, they can be combined because they share the common variable 'x'.
Constant terms like \(5, -2,\) and \(5\) are also considered like terms because they do not have any variables and can be grouped together.
Grouping like terms helps in organizing the expression for further simplification.

Combining Like Terms

Once like terms are identified, the next step in simplifying an algebraic expression is combining them. This involves adding or subtracting the coefficients of these terms.
When combining the terms in the expression

• \(1x, 4x, -22x, -1x\)

we focus on their coefficients: \(1, 4, -22,\) and \(-1\).

• Performing the operations: \(1 + 4 - 22 - 1\) yields \(-18\).

So, these terms can be combined to form \(-18x\).
Similarly, the constants \(5 - 2 + 5\) result in \(8\), when combined.
This results in a much simpler expression, \(-18x + 8\).
Combining like terms reduces the complexity of expressions, making them easier to work with.

Coefficient

The term coefficient is integral to understanding how to work with like terms. A coefficient is the numerical part of a term that is multiplied by the variable. In the terms

• \(1x, 4x, -22x,\) and \(-1x\),

coefficients are \(1, 4, -22,\) and \(-1\) respectively.
To combine like terms, you add or subtract these coefficients while keeping the variable part unchanged.

It's important to note that if a term is just \(x\), it actually has a coefficient of \(1\), and if it's \(-x\) it has a coefficient of \(-1\).
Understanding coefficients is essential for any algebraic manipulation, including simplifying expressions, solving equations, and understanding polynomials.