Problem 41 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 1 \times 2+8 x-9-1 x 2+11 x+18 $$

Step-by-Step Solution

Answer

The simplified expression is \(19x + 9\).

Step 1: Distribute and Simplify Expression

The given expression is \(1 \times 2 + 8x - 9 - 1 \times 2 + 11x + 18\). Start by simplifying simple multiplications. Evaluate \(1 \times 2\) and \(-1 \times 2\). This simplifies to \(2 + 8x - 9 - 2 + 11x + 18\).

Step 2: Combine Like Terms

Now that all constants and coefficients are dealt with, combine the like terms. For the constants, combine \(2 - 9 - 2 + 18\). For the \(x\) terms, combine \(8x + 11x\).

Step 3: Sum Constants and Coefficients

Calculating the constants: \(2 - 9 - 2 + 18 = 9\). For the \(x\) terms: \(8x + 11x = 19x\).

Step 4: Write the Simplified Expression

Combine the results from the previous step to write the simplified expression: \(19x + 9\).

Key Concepts

Polynomial Expressions

When dealing with algebra, you often encounter polynomial expressions. These expressions are composed of variables such as \(x\) and coefficients, which can be numbers like 8 or 11. They are combined using addition, subtraction, and sometimes multiplication. Polynomials are classified based on their degree, which is determined by the highest power of the variable present. In simple terms, a polynomial expression is a mathematical sentence that can contain terms like \(8x\), \(-9\), or \(11x\).

Understanding polynomial expressions is crucial as they form the basis of many algebraic operations. For instance, they might appear complex due to the arrangement of their terms, but once you recognize the "like terms," simplifying them becomes easier.

Combining Like Terms

One key aspect of simplifying polynomial expressions is combining like terms. Like terms refer to terms that contain the same variable raised to the same power, even if they have different coefficients. In our exercise, like terms such as \(8x\) and \(11x\) are identified and then combined.

Here's how you combine them:

• First, group the like terms together. For example, \(8x + 11x\) are like terms.
• Add their coefficients, in this case, \(8+11\), resulting in \(19x\).
• Apply the same method for the constant terms: \(2 - 9 - 2 + 18\).
• The total for the constants becomes \(9\).

Combining like terms simplifies expressions, making them easier to understand and solve.

Distributive Property

The distributive property helps in simplifying algebraic expressions by distributing multiplication over additions or subtractions. It states that \(a(b + c) = ab + ac\). In our exercise, this property is visible when simplifying terms like \(1 \times 2\) and \(-1 \times 2\).

To apply the distributive property effectively:

• Multiply the term outside the parentheses by each term inside.
• For \(1 \times 2\), simply compute to get 2.
• For \(-1 \times 2\), compute to get \(-2\).

By using the distributive property, you break down and simplify expressions, paving the way for further simplification, like combining like terms.