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Chapter 7: Problem 120

In the following exercises, simplify using the distributive property. $$-(3 x-7)$$

Short Answer

Expert verified
-3x + 7

Step by step solution

01

Understand the Distributive Property

The distributive property states that for any numbers or variables, \(a(b + c) = ab + ac\). It can also be written as \(a(b - c) = ab - ac\). In this problem, we are distributing a negative sign (which is the same as -1).
02

Rewrite the Expression for Clarity

Rewrite the expression \(-(3x - 7)\) as \(-1(3x - 7)\) to make it clearer that we need to distribute -1.
03

Apply the Distributive Property

Distribute -1 to both terms inside the parentheses: \(-1 \times 3x = -3x\) and \(-1 \times -7 = 7\).
04

Combine the Results

Combine the distributed terms to get the simplified expression: \(-3x + 7\).

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Key Concepts

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

simplifying expressions
Simplifying expressions is a crucial skill in algebra. It involves combining like terms and applying algebraic rules to make an algebraic expression as simple as possible.
Here are some steps:
  • Identify and combine like terms.
  • Use the distributive property to eliminate parentheses.
  • Simplify the constants if possible.
In our example, the expression \( -(3x - 7) \) has been simplified step-by-step using the distributive property.
We start by understanding that the negative sign outside the parentheses needs to be distributed to each term inside.
negative sign distribution
Distributing a negative sign may seem tricky at first, but it's quite straightforward once you understand the rule.
The negative sign in algebra can be represented as -1.
  • When distributing \( -1(3x - 7) \), it means you multiply -1 with 3x and -1 with -7.
  • First, \( -1 \times 3x = -3x \).
  • Next, \( -1 \times -7 = +7 \).
The key point to remember is that multiplying a negative by a positive gives a negative result, and multiplying a negative by a negative gives a positive result.
Finally, combining these results gives the simplified expression \( -3x + 7 \).
basic algebra rules
Mastering basic algebra rules forms the foundation for more advanced topics. Here are some essential rules to remember:
  • The distributive property: \( a(b + c) = ab + ac \).
  • Combining like terms: \( 2x + 3x = 5x \).
  • The rules of signs: \( (+) \times (+) = + \), \( (-) \times (+) = - \), \( (-) \times (-) = + \).
In our given problem, you applied these basic rules to simplify the expression.
Understanding and practicing these rules can make algebra much simpler and more approachable.

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Most popular questions from this chapter

In the following exercises, simplity using the distributive property. $$ \frac{1}{3}(15 n-6) $$

In the following exercises, convert the Fahrenheit to degrees Celsius. Round to the nearest tenth. $$ 95^{\circ} \mathrm{F} $$

For the number \(\frac{2}{5}\) find the (a) additive inverse (b) multiplicative inverse.

In the following exercises, evaluate using the distributive property. If \(d=14,\) evaluate $$ (a)-100(0.1 d+0.35) $$ $$ \begin{array}{l}{\text { (b) }-100 \cdot(0.1 d)+(-100)(0.35)} \\ {\text { to } \operatorname{show}} \\ {-100(0.1 d+0.35)=-100 \cdot(0.1 d)+(-100)(0.35)}\end{array} $$

In the following exercises, evaluate using the distributive property. If \(n=\frac{7}{8}, \quad\) evaluate $$ (a)8\left(n+\frac{1}{4}\right) $$ $$ \begin{array}{l}{\text { (b) } 8 \cdot n+8 \cdot \frac{1}{4} \quad \text { to show that }} \\ {8\left(n+\frac{1}{4}\right)=8 \cdot n+8 \cdot \frac{1}{4}}\end{array} $$
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