Problem 83 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-5 x-10 y-3 x+13 y$$

Step-by-Step Solution

Answer

The simplified form of the expression is \( -8x + 3y \)

Step 1: Identify Like Terms

From the expression \( -5 x-10 y-3 x+13 y \), we can group the terms containing the same variables together. The terms that contain the variable \( x \) are \( -5x \) and \( -3x \). The terms containing the variable \( y \) are \( -10y \) and \( 13y \)

Step 2: Combine Like Terms

Now add or subtract the coefficients of the like terms. For the \( x \) terms: \( -5x - 3x = -8x \). And for the \( y \) terms: \( -10y + 13y = 3y \)

Step 3: Put Together

The final step is to combine our results from the second step. This gives: \( -8x + 3y \)

Key Concepts

Like Terms

Like terms in algebra are terms that contain the same variables raised to the same powers. These terms can be combined through addition or subtraction. For example, in the expression

• -5x
• -3x

both contain the variable x, so they are like terms. Likewise, the terms

• -10y
• 13y

are like terms as they both contain the variable y.
Only terms that qualify as like terms can be combined. Organizing terms based on their variables is a key step in simplifying algebraic expressions.

Coefficients

Coefficients are the numerical part that is multiplied by the variable in an algebraic term. In the expression -5x, the number -5 is the coefficient, indicating it multiplies the variable x.
Understanding coefficients is essential because when simplifying expressions, you only perform operations on the coefficients of like terms. For instance:

• For the like terms -5x and -3x, you combine them by adding their coefficients: -5 + (-3) = -8;
• The variables remain the same, so the term simplifies to -8x.

This systematic approach helps in managing complex expressions more easily.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols, but does not include an equal sign. The expression

• -5x - 10y - 3x + 13y

is a simple example of this.
Each part separated by + or - signs is called a term. Algebraic expressions can be simplified by identifying and combining like terms.
This simplification process makes them easier to evaluate when substituting values for variables. It also helps in solving equations that involve these expressions.

Variables

Variables in algebra are symbols, often letters, used to represent numbers. They can hold different values, making them fundamental in creating general formulas and expressions.
In the expression -5x - 10y - 3x + 13y, x and y are variables. They act as placeholders that allow expressions to represent multiple situations with different values.
Understanding how variables function is key to solving algebraic equations and manipulating expressions. By shifting focus from specific numbers to variables, you gain the flexibility to tackle a wide range of mathematical problems.