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Chapter 2: Problem 64

Solve and check. Label any contradictions or identities. $$13-(2 c+2)=2(c+2)+3 c$$

Short Answer

Expert verified
The solution is \(c = 1\). Verification shows \(9 = 9\).

Step by step solution

01

Distribute the Right-Hand Side

Expand the right-hand side of the equation by distributing the 2 over \(c+2\). This results in \(13 - (2c + 2) = 2c + 4 + 3c\).
02

Simplify the Equation

Combine like terms on the right-hand side: \(2c + 4 + 3c\) becomes \(5c + 4\). Now the equation is \(13 - (2c + 2) = 5c + 4\).
03

Distribute on the Left-Hand Side

Distribute the negative sign on the left-hand side: \(13 - 2c - 2 = 5c + 4\). Simplify further by combining constants: \(11 - 2c = 5c + 4\).
04

Isolate the Variable

Add \(2c\) to both sides to get all the \(c\) terms on one side: \(11 = 7c + 4\). Now subtract 4 from both sides: \(7 = 7c\).
05

Solve for the Variable

Divide both sides by 7 to isolate \(c\): \(c = 1\).
06

Check the Solution

Substitute \(c = 1\) back into the original equation to verify: \(13 - (2 \cdot 1 + 2) = 2(1 + 2) + 3 \cdot 1\). Simplify both sides: \(13 - 4 = 6 + 3\), which simplifies to \(9 = 9\). The solution is verified.

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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 key skill in solving equations, helping to simplify expressions. It states that you can multiply a single term outside the parentheses by each term inside the parentheses.

For example, in the equation \(13 - (2c + 2) = 2(c + 2) + 3c\), we start by distributing the 2 on the right-hand side:

2(c + 2) becomes \(2c + 4\).

This results in the equation \(13 - (2c + 2) = 2c + 4 + 3c\). This step is essential before moving forward as it simplifies the expression, making it easier to handle.
Combining Like Terms
Combining like terms is another crucial step in simplifying equations. This means adding or subtracting terms that have identical variable parts.

In our example, after distributing, we have:

\(13 - (2c + 2) = 2c + 4 + 3c\).

Next, we combine the like terms on the right-hand side.

The terms \(2c\) and \(3c\) are like terms, giving us \(5c\). Now our equation looks like:

\(13 - (2c + 2) = 5c + 4\).

This makes the equation neater and straightforward to solve.
Isolating Variables
Isolating the variable is the process of manipulating the equation so that the variable we're solving for is by itself on one side of the equation.

Our equation now is:

\(13 - 2c - 2 = 5c + 4\).

First, simplify the left side by combining constants:

\(13 - 2 = 11\), which gives us:

\(11 - 2c = 5c + 4\).

To isolate \(c\), add \(2c\) to both sides:

\(11 = 7c + 4\).

Then, subtract 4 from both sides:

\(7 = 7c\).

Finally, divide both sides by 7:

\(c = 1\).

This step brings us to the solution for \(c\).
Verification of Solution
After finding the solution, it's essential to verify it by plugging it back into the original equation.

Substitute \(c = 1\) into the equation:

\(13 - (2 \times 1 + 2) = 2(1 + 2) + 3 \times 1\).

Simplify both sides:

\(13 - 4 = 6 + 3\).

So, we get:

\(9 = 9\).

Since both sides are equal, the solution \(c = 1\) is verified. This step ensures our solution is correct and the equation holds true for the given value of \(c\).

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Most popular questions from this chapter

In exchange for opening a new credit account, Macy's Department Stores 'subtracts \(10 \%\) from all purchases made on the day the account is established. Julio is opening an account and has a coupon for which he receives \(10 \%\) off the first day's reduced price of a camera. If Julio's final price is \(\$ 77.75,\) what was the price of the camera before the two discounts?

Abraham Lincoln's 1863 Gettysburg Address refers to the year 1776 as "four score and seven years ago." Determine what a score is.

Use an inequality and the five-step process to solve each problem. The U.S. Postal Service defines a "package" as a parcel for which the sum of the length and the girth is less than 84 in. (Length is the longest side of a package and girth is the distance around the other two sides of the package.) A box has a fixed girth of 29 in. Determine (in terms of an inequality) those lengths for which the box is considered a "package." (THE IMAGE CANNOT BE COPY)

A storekeeper goes to the bank to get \(\$ 10\) worth of change. She requests twice as many quarters as half dollars, twice as many dimes as quarters, three times as many nickels as dimes, and no pennies or dollars. How many of each coin did the storekeeper get?

Use an inequality and the five-step process to solve each problem. The formula \(R=-0.0065 t+4.3259\) can be used to predict the world record, in minutes, for the 1 -mi run \(t\) years after \(1900 .\) Determine (in terms of an inequality) those years for which the world record will be less than \(3.6 \mathrm{min}\)
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