Problem 94 - Elementary Algebra

Question

Problem: Simplify: \(12 x+15 y+3 x+2 y\) Incorrect Answer: \(12 x+15 y+3 x+2 y\) $$ \begin{aligned} &=15 x+17 y \\ &=32 x y \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression is \(15x + 17y\).

Step 1: Identify Like Terms

First, identify the like terms in the expression. Like terms are terms that have the same variable raised to the same power. In this expression, the like terms are: \(12x\) and \(3x\) \(15y\) and \(2y\)

Step 2: Combine Like Terms for 'x'

Add the coefficients of the like terms involving \(x\): \(12x + 3x = (12+3)x = 15x\)

Step 3: Combine Like Terms for 'y'

Add the coefficients of the like terms involving \(y\): \(15y + 2y = (15+2)y = 17y\)

Step 4: Write the Simplified Expression

Combine the results from Step 2 and Step 3 to write the simplified expression: \(15x + 17y\)

Key Concepts

Combining Like Terms

To simplify an algebraic expression, you first need to combine like terms. These are terms that contain the same variable raised to the same power. For instance, in the expression \(12x + 15y + 3x + 2y\), you can see a mix of terms with variables \(x\) and \(y\).

Like terms for \(x\) are \(12x\) and \(3x\), because they both have the variable \(x\). Similarly, like terms for \(y\) are \(15y\) and \(2y\).

Combining like terms simplifies the expression by adding or subtracting their coefficients. This means we perform addition or subtraction on the numbers in front of the variables (which are called coefficients), while keeping the variable part unchanged.

Coefficients in Algebra

Coefficients are the numerical parts of a term. In the term \(12x\), for example, \(12\) is the coefficient and \(x\) is the variable.

When combining like terms, you'll add or subtract the coefficients. For example, in our given problem, to combine \(12x\) and \(3x\), you keep the variable \(x\) the same and add the coefficients of 12 and 3. This results in \(15x\).

Likewise, combining \(15y\) and \(2y\) involves adding the coefficients 15 and 2, resulting in \(17y\). This helps in simplifying the expression while maintaining the correct algebraic terms.

Algebraic Simplification Techniques

Simplifying algebraic expressions involves a few key techniques:

• Identifying Like Terms: Always look for terms with the same variable and exponent. In \(12x + 15y + 3x + 2y\), for example, terms with \(x\) should be grouped together, and terms with \(y\) should be grouped together.
• Combining Like Terms: Add or subtract the coefficients of like terms. This reduces the number of terms in the expression without changing its value.
• Writing the Simplified Expression: Once you've combined all like terms, rewrite the expression in its simplest form. For our problem, this final step gives us \(15x + 17y\).

These simple steps can make even complex algebraic expressions easier to handle and understand.