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Chapter 3: Problem 70

If a number is added to 3 more than 5 times itself, the result is \(30 .\) Find the number.

Short Answer

Expert verified
The number is 4.5.

Step by step solution

01

Define the variable

Let the unknown number be represented by the variable x.
02

Formulate the equation

According to the problem, if the number x is added to 3 more than 5 times itself, the result is 30. This translates to the equation: \[ x + (5x + 3) = 30 \]
03

Simplify the equation

Combine like terms to simplify the equation: \[ x + 5x + 3 = 30 \] \[ 6x + 3 = 30 \]
04

Isolate the variable

Subtract 3 from both sides to isolate the term with the variable x: \[ 6x + 3 - 3 = 30 - 3 \] \[ 6x = 27 \]
05

Solve for x

Divide both sides of the equation by 6 to solve for x: \[ x = \frac{27}{6} \] \[ x = 4.5 \]

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Key Concepts

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

variable definition
In mathematics, a variable is a symbol used to represent an unknown value. It often appears as a letter like \(x\), \(y\), or \(z\). In the given exercise, the unknown number is represented by \(x\). Defining the variable is the first step in solving an equation since it helps us transform a word problem into a mathematical expression. For example, if we say 'a number added to 3 more than 5 times itself,' we use \(x\) to stand in for 'a number.' This makes it easier to work with and solve.
equation formulation
Formulating the equation involves translating the word problem into a mathematical statement. According to the problem: 'If a number is added to 3 more than 5 times itself, the result is 30.'
We already defined that number as \(x\). Next, we need to convert this statement into an equation. '5 times itself' means \(5x\) and '3 more than 5 times itself' means \(5x + 3\). Finally, adding the number \(x\) to \(5x + 3\) gives us:
$$x + (5x + 3) = 30$$
This equation is now ready to be solved.
isolating variable
To find the value of \(x\), we need to isolate it on one side of the equation. This involves simplifying the equation step by step.
First, combine like terms. In our equation, \(x + 5x\) results in \(6x\), giving us:
$$6x + 3 = 30$$
Next, we need to get rid of the constant term (3) sitting with \(6x\). We do this by subtracting 3 from both sides:
$$6x + 3 - 3 = 30 - 3$$
This simplifies to:
$$6x = 27$$
Our variable \(x\) is now almost isolated.
combining like terms
Combining like terms means to simplify expressions by adding or subtracting terms that have the same variable. This step is crucial for making the equation easier to solve. In our problem:
$$x + 5x + 3 = 30$$
We have \(x\) and \(5x\), which are like terms because they both contain the variable \(x\). Adding them together gives us \(6x\). So the equation becomes:
$$6x + 3 = 30$$
By combining these like terms, we simplify the equation and make it easier to isolate the variable and solve for \(x\).

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