Problem 7 - Elementary Algebra

Question

If nine more than twice a number is forty-six, what is the number?

Step-by-Step Solution

Answer

Answer: The number is 18.5.

Step 1: Define the variable

Let x be the number we are looking for.

Step 2: Write the equation

According to the given condition, nine more than twice the number is forty-six. In terms of equation, it can be written as: 2x + 9 = 46

Step 3: Solve the equation

We will now solve the equation to find the value of x: 2x + 9 - 9 = 46 - 9 \\ 2x = 37 \\ x = \frac{37}{2} \\ x = 18.5 So, the number is 18.5.

Key Concepts

Equations

In mathematics, equations are statements that assert the equality of two expressions. They are like puzzles we need to solve to find out the unknown values. For example, in the equation \(2x + 9 = 46\), there are two expressions, \(2x + 9\) and \(46\), that are stated to be equal. Equations often involve numbers, variables, and operations like addition, subtraction, multiplication, or division.

When working with equations, our goal is to figure out what numbers replace the variables to make the equation true. It's like solving a mystery by following clues to find out which numbers fulfill the statement in the equation.

Understanding how to manipulate and rearrange equations is crucial. It's a core skill in algebra that allows you to tackle more complex problems in math.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. Linear equations can always be written in the standard form \(ax + b = c\), where \(x\) is the variable, and \(a\), \(b\), \(c\) are constants. Our task is to solve these equations by finding the value of \(x\).

There are a few key steps that help in solving these equations easily:

• Isolate the variable: Get \(x\) alone on one side of the equation.
• Perform inverse operations: If something is being added to \(x\), subtract it from both sides. If \(x\) is multiplied by a number, divide both sides by that number.
• Simplify: Reduce the equation step by step until the variable has a coefficient of 1.

In our example, we had the equation \(2x + 9 = 46\). By subtracting 9 from both sides, we got \(2x = 37\). Then, dividing both sides by 2, we found \(x = 18.5\).

These steps can be applied to any linear equation to find the unknown value.

Variables

Variables in algebra are symbols used to represent unknown numbers. Most commonly, letters like \(x\), \(y\), or \(z\) are used as variables. A variable acts as a placeholder for numbers in equations and expressions and allows flexibility to solve various problems without specifying exact values.

By using variables, we can create general equations, like \(2x + 9 = 46\), that can model real-world situations. Variables help in forming mathematical relationships between different quantities. They are essential tools not just in mathematics, but also in science and engineering where modeling and simulations are necessary.

Understanding the role of variables is crucial because they help simplify problems, enable us to generalize solutions, and are foundational in developing more advanced mathematical concepts. Working with variables enhances problem-solving skills by helping us think abstractly and solve real-world problems efficiently.