Problem 74 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(8 p+6-3(3 p-1)\)

Step-by-Step Solution

Answer

-p + 9

Step 1 - Distribute the multiplication

Distribute the -3 across the terms inside the parentheses: -3(3p - 1) = -3 × 3p + (-3) × -1, resulting in -9p + 3.

Step 2 - Rewrite the expression

Substitute the distributed terms back into the original expression: 8p + 6 - 9p + 3.

Step 3 - Combine like terms

Combine the like terms: 8p and -9p as well as 6 and 3: 8p - 9p + 6 + 3 which simplifies to -p + 9.

Key Concepts

Distributive Property

The Distributive Property is a useful tool in algebra for simplifying expressions. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. This is expressed as: \(a(b + c) = ab + ac\).
In our exercise, we applied the distributive property to the term: \(-3(3p - 1)\). This means we multiply \(-3\) by each term inside the parentheses.
Here’s the breakdown: \-3 \times 3p = -9p\ and \-3 \times -1 = 3\.
Now, the expression becomes: \(-9p + 3\). This is how we used the distributive property to simplify the expression initially.

Combining Like Terms

Combining like terms is a critical step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. Only the coefficients (numbers in front of the variables) can be added or subtracted.
In the exercise, the expression after distribution was: \8p + 6 - 9p + 3\. Here, we have two sets of like terms: \(8p\) and \(-9p\), as well as \(6\) and \(3\).
Combining like terms means adding the coefficients of \8p \) and \(-9p)\) to get \(8p - 9p = -p\). Similarly, add \(6\) and \(3\) to get \(6 + 3 = 9\). The simplified form now is: \(-p + 9)\.

Linear Equations

A linear equation is an equation between two variables that gives a straight line when plotted on a graph. The standard form is \(ax + b = 0\). What we simplified in our exercise was not a full equation but an algebraic expression.
To turn an expression into an equation, it needs to be set equal to something (like zero). An example of a linear equation based on our simplified expression \(-p + 9\) would be: \(-p + 9 = 0\).
Solving this would give you the value of \(p\). Linear equations are foundational for understanding more complex algebraic concepts.