Problem 37 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 21 y-15 x+40 x y-6-11 y+7-12 x-x y $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(21y - 15x + 40xy - 6 - 11y + 7 - 12x - xy\). Answer: \(10y - 27x + 40xy + 1 - xy\)

Step 1: Identify like terms

In the given expression, we can find like terms by identifying which terms have the same variables raised to the same powers. For the expression: $$ 21y - 15x + 40xy - 6 - 11y + 7 - 12x - xy $$ We can group the terms as follows: - Terms involving x: \(-15x - 12x-xy\) - Terms involving y: \(21y - 11y\) - Terms involving xy: \(40xy\) - Constants: \(-6 + 7\)

Step 2: Combine like terms

Next, we can combine each group of like terms by performing addition or subtraction operations as necessary. For terms involving x: $$ -15x - 12x - xy = (-15 - 12)x - xy = -27x - xy $$ For terms involving y: $$ 21y - 11y = (21 - 11)y = 10y $$ Constants: $$ -6 + 7 = 1 $$

Step 3: Rewrite the expression with simplified terms

After combining like terms in each category, we can rewrite the expression as follows: $$ 21y - 15x + 40xy - 6 - 11y + 7 - 12x - xy = 10y - 27x + 40xy + 1 - xy $$ This simplified expression is the final answer: $$ 10y - 27x + 40xy + 1 - xy $$

Key Concepts

Like Terms

Understanding like terms is key in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. They might look different at first glance, but by identifying their common elements, we can unify them into a single entity. For instance, in the expression \(-15x - 12x - xy\), \(-15x\) and \(-12x\) are like terms because they both involve the variable \(x\). They have different coefficients, which indicates how many times that specific term, in this case \(x\), is counted.

Likewise, in \(21y-11y\), both terms involve \(y\), and therefore they are like terms as well. Constants are also considered like terms with each other, like \(-6\) and \(+7\). By looking for like terms, you simplify complex expressions by grouping and combining them in the next steps. This crucial process ensures accurate simplification and a clear final expression.

Simplification

Simplification is the next step after identifying the like terms. It involves combining these terms to make the expression less complex, usually through addition or subtraction. Once you have grouped the like terms, add or subtract the coefficients of these terms to combine them.

For example, using the previously identified like terms, combine \(-15x\) and \(-12x\) to get \((-15 - 12)x = -27x\). For the terms involving \(y\), \(21y - 11y = 10y\), which means we simply perform arithmetic operations on their coefficients while keeping the variable part intact. Constants are often the easiest to combine, such as \(-6 + 7 = 1\).

Simplification reduces the expression to fewer terms and helps in clearer interpretation or further computation.

Combining Terms

Combining terms is about summing up all the consolidated components after simplification. It often results in a more efficient and compact representation of the expression. In this concluding phase, you bring together all the results from simplifying each group of like terms.

Using our example: the simplified terms were \(10y\), \(-27x\), \(40xy\), \(1\), and \(-xy\). When you combine these, maintain the order of terms based on their type, which sometimes helps in specific applications of algebra. Ensure the expression is neat and ready for any additional algebraic or application tasks.

The final expression, \(10y - 27x + 40xy + 1 - xy\), presents the combined outcome of the simplification process. It's free from repetition and offers a clear, direct form that can be used for solving equations or further algebraic manipulations.