Problem 45 - Elementary Algebra

Question

Simplify. $$ 3 x+5-2 y+7-5 x+3 y $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x + y + 12\).

Step 1: Group Like Terms

Identify and group the like terms in the expression. Like terms are terms that have the same variable raised to the same power. In our expression we have: - Terms involving 'x': \(3x\) and \(-5x\).- Terms involving 'y': \(-2y\) and \(3y\).- Constant terms (no variables): \(5\) and \(7\).

Step 2: Combine Like Terms for 'x'

Add the coefficients of the like terms that have the variable 'x':\(3x - 5x = (3-5)x = -2x\).

Step 3: Combine Like Terms for 'y'

Add the coefficients of the like terms that have the variable 'y':\(-2y + 3y = (-2+3)y = 1y = y\).

Step 4: Combine Constant Terms

Add the constant terms, which do not have any variables:\(5 + 7 = 12\).

Step 5: Write Simplified Expression

Combine the results from steps 2, 3, and 4 to write the simplified expression:The simplified expression is \(-2x + y + 12\).

Key Concepts

Like Terms

In algebra, understanding "like terms" is crucial for simplifying expressions effectively. Like terms are terms within an algebraic expression that have the same variable components. This means they must have the same variable raised to the same power. For instance, in the expression \(3x + 5 - 2y + 7 - 5x + 3y\), the terms \(3x\) and \(-5x\) are like terms because they both involve the variable \(x\), while \(-2y\) and \(3y\) are like terms due to the variable \(y\). Unlike terms cannot be combined because their variable parts are different. Here’s what to remember:

• Always check the variable and the exponent.
• If both match, the terms can be combined.

This technique of grouping and combining like terms is fundamental in algebraic simplification, making complex expressions more manageable.

Coefficients

Coefficients are numbers that are placed in front of variable terms. They essentially "scale" the variable. Understanding coefficients is important for combining like terms in algebra. For example, in the terms \(3x\) and \(-5x\), \(3\) and \(-5\) are coefficients of \(x\) respectively. They help determine the resulting value when like terms are combined. Here’s a simple way to work with coefficients:

• Identify the coefficients of like terms.
• Add or subtract them as indicated by the operation between the terms.
• Keep the variable part constant.

For instance, when you combine \(3x - 5x\), you're essentially calculating \((3 - 5)x\), which equals \(-2x\). By focusing on coefficients, you simplify the numerical component of variable terms.

Variable Terms

Variable terms in an algebraic expression like \(3x\) or \(-2y\) are parts of an equation that include variables. These terms represent numbers that can vary based on the value of the variable. A variable is typically represented by a letter, like \(x\) or \(y\).Variable terms are what you manipulate when combining like terms. Here's how they work:

• If two terms have the same variable, they can be added or subtracted from each other.
• After combining their coefficients, the variable stays the same.

For example, in combining \(-2y\) and \(3y\), since they both contain the variable \(y\), they simplify to \(y\). This reflects the core idea that variable terms with the same letter can be simplified through coefficient addition or subtraction.

Constant Terms

Constant terms are the simple, straightforward numbers in an expression that do not contain any variables. They are added or subtracted outright since there are no variables to consider. In the expression \(3x + 5 - 2y + 7 - 5x + 3y\), the numbers \(5\) and \(7\) are constant terms.To simplify constant terms, you just add or subtract them as you would with regular numbers:

• Identify all constant terms in the expression.
• Perform the indicated arithmetic operations (add or subtract).

In this case, you combine \(5\) and \(7\) to get \(12\). Simplifying constant terms helps in reducing the complexity of an expression by focusing on the variable-free part.