Problem 66 - Introductory Algebra for College Students

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. The sum of twice a number and 9 is 29

Step-by-Step Solution

Answer

The equation translated from the given sentence is \(2x + 9 = 29\).

Step 1: Identify the Unknown

Given that the letter \(x\) will represent the unknown number we are trying to find.

Step 2: Translate the Words into Mathematical Expressions

The words 'the sum of' generally signifies the operation of addition. 'Twice a number' can be represented as '2x', as the word 'twice' means 'two times'. Therefore, our sentence 'The sum of twice a number and 9' can be translated into the mathematical expression '2x + 9'.

Step 3: Complete the Equation

The phrase 'is 29' sets our expression equal to 29. Therefore, the equation translated from the original sentence is '2x + 9 = 29'.

Key Concepts

Algebraic Expressions

When we talk about algebraic expressions, we're referring to combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In the context of our example, the phrase 'twice a number and 9' is an algebraic expression. It involves multiplication ('twice' means to multiply by two) and addition (adding 9). For example, if we imagine our unknown number is 5, then 'twice this number and 9' would be 2(5) + 9. But since our number is unknown, we use a variable, typically the letter 'x', to represent it. Thus 'twice a number and 9' becomes the algebraic expression '2x + 9'.

Algebraic expressions act as a bridge between everyday language and the numerical world they represent, allowing us to articulate and manipulate quantities even when they're unknown. Math problems often start with a word problem that needs to be translated into an algebraic expression before any calculations can begin.

Solving Algebraic Equations

To solve algebraic equations like the one in our example, '2x + 9 = 29', we perform operations to isolate the variable and find its value. Here are the steps applied to this particular problem:

1. First, we look at the equation and identify what operations are being performed on our variable. In this case, '2x' is being multiplied by 2 and 9 is being added to the result.
2. We then work to reverse these operations to solve for 'x'. To start, we would subtract 9 from both sides to 'undo' the addition. This leaves us with '2x = 20'.
3. Finally, we divide both sides by 2 to 'undo' the multiplication, which gives us 'x = 10'.

Solving algebraic equations usually involves these inverse operations — subtraction neutralizes addition, division neutralizes multiplication — until the variable is left on one side of the equal sign, and the solution is on the other.

Representing Unknowns with Variables

In algebra, a variable is a symbol, usually a letter, that stands in for an unknown value. Variables are the 'placeholders' of math. They allow us to work with mathematical statements and relationships even when we don’t know all the numbers involved. In the given exercise, the variable 'x' represents an unknown number.

We choose a variable to stand in for a number that we don't know yet but are trying to find. It's like a blank space in a sentence that we fill in once we've figured out what should go there. By representing unknowns with variables, we can form equations that relate different quantities and solve them to find out unknown values. This is an essential part of algebra and helps in solving a vast array of problems, from simple equations to complex real-world applications.