Problem 58 - Beginning and Intermediate Algebra

Question

Simplify each expression. $$ \frac{1}{5} y+4-4-\frac{1}{5} y $$

Step-by-Step Solution

Answer

0

Step 1: Identify Like Terms

Examine the expression \[ \frac{1}{5} y + 4 - 4 - \frac{1}{5} y \] and identify like terms. The terms that involve the variable \( y \) are \( \frac{1}{5} y \) and \( -\frac{1}{5} y \). The constant terms are 4 and -4.

Step 2: Combine Like Terms Involving Variables

Combine the like terms involving \( y \): \[ \frac{1}{5} y - \frac{1}{5} y = 0 \] These terms cancel each other out.

Step 3: Combine Constant Terms

Combine the constant terms: \[ 4 - 4 = 0 \] These also cancel each other out.

Step 4: Write the Simplified Expression

Since both the variable terms and constant terms have canceled out, the simplified expression is \[ 0 \]

Key Concepts

Like Terms

Like terms are terms that contain the same variable raised to the same power. For example, in the expression \[ \frac{1}{5} y + 4 - 4 - \frac{1}{5} y \], the like terms involving the variable are \[ \frac{1}{5} y \] and \[ -\frac{1}{5} y \]. Both terms have the variable \[ y \] raised to the first power. Similarly, the constants 4 and -4 are also like terms because they are both numbers without any variables. Identifying like terms is the first step in simplifying algebraic expressions.

Once we identify like terms, we can work on simplifying the expression by combining these terms together.

Combining Terms

Combining terms means adding or subtracting like terms to simplify an algebraic expression. This is done by combining the coefficients of the like terms. For instance, in the expression \[ \frac{1}{5} y + 4 - 4 - \frac{1}{5} y \], we can combine \[ \frac{1}{5} y \] and \[ -\frac{1}{5} y \] because they have the same variable.

When combining these variable terms, we get:
\[ \frac{1}{5} y - \frac{1}{5} y = 0 \].
Similarly, we combine the constant terms 4 and -4: \[ 4 - 4 = 0 \]. Combining like terms helps in making the expression simpler and easier to work with.

Algebraic Simplification

Algebraic simplification is the process of reducing an algebraic expression to its simplest form. This involves combining like terms and performing any possible arithmetic operations. In the expression \[ \frac{1}{5} y + 4 - 4 - \frac{1}{5} y \], after identifying and combining the like terms, we find that all terms cancel out, leaving us with:
\[ 0 \].
\br> Simplifying an expression can make it much easier to understand and solve problems that involve the expression. It is a crucial skill in algebra that helps students to work efficiently with equations and other mathematical expressions.