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Chapter 2: Problem 23

Solve each equation, and check the solution. \(5(p-1)-p=2 p-7\)

Short Answer

Expert verified
The solution is \(p = -1\).

Step by step solution

01

Distribute the 5

Apply the distributive property to remove the parentheses. Calculate the expression as follows: \[5(p-1)-p = 5p - 5 - p.\]
02

Combine like terms

Combine the terms involving \(p\) to simplify the equation: \[5p - p - 5 = 4p - 5.\]
03

Rearrange the equation

Now, the equation is \[4p - 5 = 2p - 7.\] Subtract \(2p\) from both sides to get all terms involving \(p\) on one side: \[4p - 2p - 5 = -7 \Rightarrow 2p - 5 = -7.\]
04

Solve for \(p\)

Add 5 to both sides to isolate the term with \(p\): \[2p - 5 + 5 = -7 + 5 \Rightarrow 2p = -2.\] Divide both sides by 2: \[p = \frac{-2}{2} = -1.\]
05

Check the solution

Substitute \(p = -1\) back into the original equation to verify: \[5((-1)-1) - (-1) = 2(-1) - 7.\] Simplify both sides: \[5(-2) + 1 = -2 - 7 \Rightarrow -10 + 1 = -9 \Rightarrow -9 = -9.\] Since both sides are equal, \(p = -1\) is the correct solution.

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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 an essential concept when solving linear equations. It allows you to eliminate parentheses by distributing a multiplication over addition or subtraction inside the parentheses.

In our example equation, 5(p-1)-p=2p-7, we apply the distributive property in Step 1. To distribute the 5, multiply it by both terms inside the parentheses: \(5 \times p - 5 \times 1 = 5p - 5\).

This step is crucial because it simplifies the equation and prepares it for combining like terms in the next step.

combining like terms
Once the distributive property has been applied, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power.

In our example, once we have 5p-5-p, we combine the like terms involving p: \(5p - p = 4p\). We also consolidate any constant terms, but in this example, there are no other constants to combine at this stage.

This process simplifies the equation significantly and makes it easier to isolate the variable in the succeeding steps.
isolation of variables
The next major step is isolating the variable. This involves rearranging the equation so that the variable (in this case, p) is on one side of the equation and all constants are on the other.

In our example, we first subtract 2p from both sides to get all the p terms on one side: \(4p - 2p - 5 = -7 \rightarrow 2p - 5 = -7\). Then, we add 5 to both sides to start isolating p: \(2p - 5 + 5 = -7 + 5 \rightarrow 2p = -2\).

Finally, we divide both sides by 2: \(p = \frac{-2}{2} = -1\). This step ensures that the variable is by itself, allowing us to find its value.

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Most popular questions from this chapter

Norway earned 39 medals at the 2018 Winter Olympics. The number of gold medals earned was the same as the number of silver medals. The number of bronze medals earned was 3 fewer than the number of silver medals. How many of each kind of medal did Norway earn? (Data from The Gazette.)

Solve each equation for \(y\) $$ x-5 y=10 $$

Solve each equation for \(y\) $$ -3 x+5 y=-15 $$

Graph the solution set, and write it using interval notation \(-7 \leq \frac{5}{4} r-1 \leq-1\)

The United States earned 121 medals at the 2016 Summer Olympics. The number of silver medals earned was 1 fewer than the number of bronze medals. The number of gold medals earned was 8 more than the number of bronze medals. How many of each kind of medal did the United States earn? (Data from The World Almanac and Book of Facts.)
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