Problem 46 - Elementary Algebra

Question

Simplify. $$ -y+8 x-3+14 x+1-y $$

Step-by-Step Solution

Answer

The simplified expression is \( 22x - 2y - 2 \).

Step 1: Combine Like Terms

First, identify like terms in the expression. We have terms with \( x \), terms with \( y \), and constant terms. Specifically: \( 8x \) and \( 14x \) are like terms, \( -y \) and \( -y \) are like terms, and \( -3 \) and \( 1 \) are like terms.

Step 2: Simplify the \( x \) Terms

Combine the \( x \) terms: \( 8x + 14x = 22x \).

Step 3: Simplify the \( y \) Terms

Combine the \( y \) terms: \( -y - y = -2y \).

Step 4: Simplify the Constant Terms

Combine the constant terms: \( -3 + 1 = -2 \).

Step 5: Reassemble the Expression

After combining like terms, the simplified expression is \( 22x - 2y - 2 \).

Key Concepts

Like Terms

In algebra, a key concept is understanding what 'like terms' are. Like terms are components in an algebraic expression that have the same variable part. This means the variables have the same powers; only the coefficients can differ. For example:

• In the expression \(-y + 8x - 3 + 14x + 1 - y\), the terms \(8x\) and \(14x\) are like terms because they both involve the variable \(x\). They can be combined as they express quantities of the same algebraic component.
• Similarly, \(-y\) and \(-y\) also qualify as like terms because they have the variable \(y\) with the same power, which is 1, and the same coefficient, albeit it being a negative.
• Even constant terms, such as \(-3\) and \(1\), are considered like terms because they are plain numbers without variables.

Identifying like terms is the first crucial step toward simplifying an algebraic expression. When you find all the like terms in an expression, you can easily manipulate and simplify it.

Simplification

Simplification in algebra refers to the process of reducing an expression to its simplest form. This involves combining like terms to streamline calculations and make the expression easier to interpret.

When simplifying:

• The first step is to identify all like terms within the expression. For instance, \(8x\) and \(14x\) are like terms.
• Next, you combine these like terms. So, \(8x + 14x\) results in \(22x\).
• Repeat this for all groups of like terms. For example, combining \(-y\) and \(-y\) results in \(-2y\), while \(-3 + 1\) simplifies to \(-2\).

Subsequent steps involve rewriting the expression using these simplified components, which results in a much tidier presentation of the original formula. In our case, the simplified form is \(22x - 2y - 2\). This process not only helps in managing the expression but also in carrying out further algebraic manipulations with ease.

Combining Terms

Combining terms is a fundamental process in simplifying algebraic expressions. This task involves aggregating like terms to reduce the complexity of the expression. It often requires careful arithmetic and attention to the signs of coefficients.

When combining terms:

• Focus first on each type of variable term. For instance, handle all terms involving \(x\) like \(8x\) and \(14x\), combining them to get \(22x\).
• Next, address the terms involving another variable, in this case, \(-y\) and \(-y\), which result in \(-2y\).
• Finally, bring together any constant terms, such as \(-3\) and \(1\), leading to \(-2\).

This methodical approach ensures clarity and precision. By repeatedly combining terms, an entire expression can be reduced to its simplest and most manageable form, which is vital for solving algebraic equations efficiently. The process results in a clearer, and often shorter, expression, reflecting only the necessary components and their correct relations.