Problem 17 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. Nine times a number is 30 more than three times that number. Find the number.

Step-by-Step Solution

Answer

The number x is 5.

Step 1: Translate the word problem into an equation

The problem describes the relationships of 'nine times a number' and '30 more than three times that number'. We can encode these relationships as an equation. If we represent the unknown number as x, the equation becomes \(9x = 3x + 30\). This is because the statement 'nine times a number' translates to '9 times x or 9x' and '30 more than three times that number' translates to '3 times x plus 30 or 3x plus 30'.

Step 2: Simplify the equation and isolate the variable

Now we can simplify the equation and solve it for x. We start by subtracting \(3x\) from both sides of the equation to get \(6x = 30\). Then, we divide both sides of the equation by 6 to isolate x. The equation becomes \(x = \frac{30}{6}\).

Step 3: Solve the equation

Finally we solve the equation for x. If we calculate the fraction, we find that \(x = 5\).

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and they form the basis for more complex mathematical concepts. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable. These equations appear in the form of a straight line when graphed on a coordinate plane and generally look like \( ax + b = 0 \), where \('a'\) and \('b'\) are constants, and \('x'\) is the variable.

The equation featured in our exercise, \( 9x = 3x + 30 \), is a prime example of a linear equation. Here, \( 9x \) and \( 3x \) are terms where the variable \( x \) is raised to the first power, and \( 30 \) is a constant. The key to solving a linear equation is to isolate the variable on one side of the equation, leading us to the value that \( x \) represents. Through this process, you uncover the x-value that makes the equation true.

Variable Isolation

Variable isolation is a methodical approach to solving an equation that allows us to find the value of an unknown. The main goal of this process is to get the variable by itself on one side of the equation. To do this, we usually perform a series of inverse operations.

As we saw in the exercise, we started with the equation \( 9x = 3x + 30 \). To isolate \( x \), we first eliminate other instances of the variable from one side, which leads to \( 6x = 30 \) after subtracting \( 3x \) from both sides. Afterwards, we divide both sides by 6, the coefficient of \( x \), giving us \( x \). This leaves \( x \) by itself and equals to a number, which in our case is 5. Through variable isolation, we're able to simplify complex statements to find concrete solutions.

Word Problems in Algebra

Word problems in algebra are a way to turn real-world problems into mathematical equations that can be solved analytically. They often challenge students because they require a translation of text into a mathematical statement. The first step to solving word problems is to define the variable—what we're trying to find—and understand the relationships described in the text.

The given problem, 'Nine times a number is 30 more than three times that number,' requires us to interpret this scenario and create an equation based on it. In our exercise, we define \( x \) as 'a number' and translate the problem into the linear equation \( 9x = 3x + 30 \). The translation and setup are as critical as the algebraic solution itself. Often, drawing a diagram or listing out known and unknown elements can be helpful strategies in understanding and organizing the information before solving the equation.