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Chapter 1: Problem 74

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

Short Answer

Expert verified
-p + 9

Step by step solution

01

- Distribute the multiplication

Distribute the -3 across the terms inside the parentheses: -3(3p - 1) = -3 脳 3p + (-3) 脳 -1, resulting in -9p + 3.
02

- Rewrite the expression

Substitute the distributed terms back into the original expression: 8p + 6 - 9p + 3.
03

- 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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

Write a numerical expression for each phrase, and simplify the expression. The sum of -18 and \(-6,\) divided by the product of 2 and -4

Write each expression without parentheses. $$ -(-5 c-4 d) $$

Solve each problem. The surface, or rim, of a canyon is at altitude \(0 .\) On a hike down into the canyon, a party of hikers stops for a rest at \(130 \mathrm{~m}\) below the surface. The hikers then descend another \(54 \mathrm{~m}\). Write the new altitude as a signed number.

Solve each problem. In \(2000,\) the U.S. federal budget had a surplus of \(\$ 236\) billion. In \(2016,\) the federal budget had a deficit of \(\$ 616\) billion. Express the difference between these amounts as a positive number. (Data from U.S. Office of Management and Budget.)

Use the distributive property to rewrite each expression. $$ -11(x+4) $$
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