Problem 107 - Intermediate Algebra

Question

Simplify each expression. $$ -5.7 p t-p+5.1 p t+12 p $$

Step-by-Step Solution

Answer

The simplified expression is \[-0.6pt + 11p\].

Step 1: Identify Like Terms

In the expression \[-5.7pt - p + 5.1pt + 12p\],identify the terms that can be combined. 'Like terms' are terms that have the exact same variables raised to the same powers. Here, the terms that include 'pt' are -5.7pt and 5.1pt, while terms that include 'p' are -p and 12p.

Step 2: Combine Like Terms with 'pt'

Add together the like terms that include 'pt':\[-5.7pt + 5.1pt = (-5.7 + 5.1)pt = -0.6pt\].

Step 3: Combine Like Terms with 'p'

Next, add the like terms that include 'p':\[-p + 12p = (-1 + 12)p = 11p\].

Step 4: Write the Simplified Expression

Combine the results from Steps 2 and 3 to write the simplified expression. The expression \[-5.7pt - p + 5.1pt + 12p\] simplifies to \[-0.6pt + 11p\].

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions is like cleaning up clutter to make solving problems easier. When you simplify an expression like \(-5.7pt - p + 5.1pt + 12p\), you focus on combining terms that are alike. "Like terms" have identical variables, and their variables are raised to the same power.

This makes simplification a much simpler task! In our example, first spot the like terms: -5.7pt with 5.1pt, and -p with 12p. Then, add or subtract these terms separately. This step-by-step process allows a messy string of numbers and letters to become a tidy expression, like our simplified \(-0.6pt + 11p\).

Following these simplification steps can significantly increase your efficiency when working through algebraic problems. It reduces error chances and ensures accuracy in larger equations.

Algebraic Expressions

Algebraic expressions are like puzzles made of numbers, variables, and math operations. Each piece has its own role, preparing you for more complex math tasks. Understanding algebraic expressions is crucial as it lays the groundwork for manipulating and solving equations.

An expression like \(-5.7pt - p + 5.1pt + 12p\) is built from:

• Constants, -5.7 and 5.1
• Variables, p and t
• Coefficients, such as -1 and 12, showing how much each variable counts.

Grasping these parts helps you see the relationships within the expression. It also makes recognizing opportunities to simplify and solve easier. Remember, algebraic expressions don't have equal signs—that's for equations. Here, the goal is to streamline, clarify, and prepare for the next steps in algebra.

Intermediate Algebra

Intermediate algebra builds on the basics by introducing more intricate tasks, such as handling multiple variables and combining terms with fractional or decimal coefficients. As seen in the expression \(-5.7pt - p + 5.1pt + 12p\), you deal with decimals and negative signs.

Mastering intermediate algebra means:

• Becoming proficient in recognizing and combining like terms.
• Handling complex coefficients with confidence.
• Simplifying expressions as a routine part of problem-solving.

This level of algebra is about applying your foundation to new challenges. It's about seeing problems from a higher level, simplifying with precision, and preparing for advanced topics like functions and quadratic equations.

Intermediate algebra hones your skills to the point where expressions become second nature, and you're equipped to tackle more advanced mathematical concepts.