Problem 10 - 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. The sum of five and three times a number is \(59 .\) Find the number.

Step-by-Step Solution

Answer

The number represented by \(x\) is 18.

Step 1: Formulate the problem as an equation

Translate the words of the problem into a mathematical equation. Here, 'the sum of five and three times a number' can be represented as \(5 + 3x\), where \(x\) is the unknown number. This sum is given to be equal to \(59\), so the entire equation can be written as \(5 + 3x = 59 .\)

Step 2: Simplify the equation

When solving for \(x\), the goal is to isolate \(x\) on one side of the equation. Begin by subtracting 5 from both sides of the equation to get rid of the 5 on the left side of the equation and simplify. This results in \(3x = 59 - 5\), which simplifies to \(3x = 54 .\)

Step 3: Solve the equation for x

Then, divide each side by 3 in order to solve for \(x\). This gives: \(x = 54 / 3\). This simplifies to \(x = 18 .\)

Key Concepts

Variables

In algebra, a variable is a crucial concept that acts as a placeholder for unknown values. It's like a mystery waiting to be solved. Variables are often represented by letters such as \(x\), \(y\), or \(z\). In our exercise, the problem introduced a number that wasn't given directly, so we used the letter \(x\) to stand in its place.

When dealing with equations, variables allow us to formulate problems based on conditions mentioned in the exercise. For instance, \'three times a number\' translates to \(3x\), where \(x\) is our variable. Understanding this helps to translate verbal expressions into mathematical equations more easily.

Remember that variables can take on any value that makes the equation true. Solving an equation helps us determine what specific number a variable should represent.

Equation Solving

Equation solving in algebra involves finding the value of the unknown variable that satisfies the given mathematical statement. Let's walk through the solution to our problem.

Initially, we translated the verbal statement into the equation form: \(5 + 3x = 59\). This equation states that when you add 5 to three times a number \(x\), you get 59.

The next step in solving the equation is isolating the variable. We did this by subtracting 5 from both sides, simplifying the equation to \(3x = 54\). Finally, to solve for \(x\), we divided both sides by 3, yielding \(x = 18\). This means that the number is 18.

When solving equations, structure your solution like a balanced scale. Whatever you do to one side, do to the other to keep the equation balanced.

Simplifying Expressions

Simplifying expressions is a key step in making equations easier to solve. The primary goal is to reduce an equation to its simplest form, which helps us isolate the variable.

In our equation \(5 + 3x = 59\), we simplified by removing unnecessary terms. Subtraction of 5 from both sides gave us \(3x = 54\). This step made it easier to isolate \(x\) by performing further operations.

Some tips for simplifying expressions include:

• Perform arithmetic operations as needed, like addition or subtraction.
• Consolidate terms that are alike or that can be combined.
• Always aim for the expression to have the variable on one side and constants on the other.

Simplification turns complex equations into manageable ones, paving the way for clearer solutions. By mastering this step, you ensure a smoother path to finding answers in algebra.