Problem 73 - Elementary Algebra

Question

Simplify. $$ 12 y-34 x-(23 y-15 x) $$

Step-by-Step Solution

Answer

The simplified expression is \(-11y - 19x\).

Step 1: Distribute the Negative Sign

The expression given is \(12y - 34x - (23y - 15x)\). The first step is to distribute the negative sign inside the parentheses. This means changing the signs of each term within the parentheses:\[12y - 34x - 23y + 15x\]

Step 2: Combine Like Terms for \(y\)

You now have the terms with \(y\): \(12y\) and \(-23y\). Combine these like terms by subtracting their coefficients:\[12y - 23y = -11y\]

Step 3: Combine Like Terms for \(x\)

Next, consider the terms with \(x\): \(-34x\) and \(+15x\). Combine these by adding their coefficients:\[-34x + 15x = -19x\]

Step 4: Write the Final Simplified Expression

Combine the results from Step 2 and Step 3 to write the final simplified expression:\[-11y - 19x\]

Key Concepts

Distribution

Distribution is a key algebraic rule used to simplify expressions. It involves spreading out or distributing a term across terms inside parentheses. In our exercise, we dealt with distributing a negative sign.
For example, when you have an expression like \[-(23y - 15x)\], you treat the negative sign as -1, multiplying it through the terms inside the parentheses.

• This action flips the signs: turning \[23y\] into \[-23y\], and \[-15x\] into \[+15x\].
• Remember, distributing is akin to sharing each term on an equal basis.

This crucial step ensures that you do not overlook the negative sign's impact, keeping your simplification accurate.

Combining Like Terms

Combining like terms focuses on terms with similar variables and exponents. This step condenses expressions by adding or subtracting coefficients of these terms. In our solved problem, like terms include both variable 'y' terms and variable 'x' terms.

• \[12y\] and \[-23y\] are combined to become \[-11y\], simply by subtracting and adding the coefficients.
• Similarly, \[-34x\] and \[+15x\] combine to form \[-19x\].

Always make sure to match terms correctly, checking their variables and powers to combine them accurately.
This practice will simplify and streamline algebraic expressions, leading to more manageable forms.

Simplifying Expressions

Simplifying expressions is the process of rewriting complex expressions in a simpler or more efficient manner. The goal is to achieve a form that is both easier to read and work with.
The combination of distribution and combining like terms in our exercise turned the original expression: \[12y - 34x - (23y - 15x)\] into its simplest form: \[-11y - 19x\]. The simplification process involves:

• Applying distribution correctly, ensuring you adjust all necessary signs.
• Accurately combining like terms to manage each variable methodically.

By mastering these steps, you can handle larger and more complex algebraic tasks, making simplifications clearer and more efficient in subsequent mathematical problems.