Problem 44 - Algebra A Combined Function

Question

Rewrite each sentence using mathematical symbols. Do not solve the equations. Twice a number, added to \(6,\) is 3 less than the number.

Step-by-Step Solution

Answer

The equation is \( 2x + 6 = x - 3 \).

Step 1: Define the variable

Let's denote the unknown number by the mathematical symbol \( x \).

Step 2: Express 'Twice a number' mathematically

The phrase 'twice a number' means two times the number, which can be written as \( 2x \).

Step 3: Add 6 to twice the number

Add \( 6 \) to the expression for twice a number, resulting in \( 2x + 6 \).

Step 4: Express '3 less than the number' mathematically

The phrase '3 less than the number' can be mathematically represented as \( x - 3 \).

Step 5: Set up the equation

According to the problem, 'twice a number, added to 6, is 3 less than the number', which can be expressed as the equation \( 2x + 6 = x - 3 \).

Key Concepts

Mathematical Symbols

Mathematical symbols are like a universal language that help us express complex ideas clearly and concisely. In algebra, they allow us to communicate the relationships between numbers and functions systematically. For example, the symbol "+" signifies addition, while "=" indicates equality. When we deal with algebraic expressions, using these symbols helps us describe what happens to numbers in an efficient way.

Consider how we can use symbols to rewrite real-world situations. Instead of writing "twice a number, added to 6," we might express this simply as \(2x + 6\)." Here, "2" is indicating multiplication with a number \(x\), and the "+" sign instructs us to add 6.

By learning to convert phrases into mathematical symbols, we can transform everyday language into a format that can be easily analyzed and solved mathematically.

Variables in Algebra

Variables in algebra are symbols, usually letters, that stand in for unknown or unspecified numbers. When we say "a number" in a problem, we often use a variable like \(x\) to represent it. This is because we might not know the exact value yet or might want to express the relationship generally. Using variables makes algebra powerful, as it lets us write equations that apply to many situations.

In the exercise, we use the variable \(x\) to denote an unknown number. This turns the sentence into an expression that can be manipulated: \(2x + 6 = x - 3\). It's essential to carefully define what each variable represents. Here, it represents the number we are trying to determine from the context of the problem.

The neat thing about using variables is that they allow us to handle equations and functions abstractly, discover patterns, and find solutions in a general manner.

Equations

Equations are mathematical statements that assert the equality of two expressions. When we talk about equations in algebra, we're usually trying to find the value of the unknown variable. An equation will have an "=" sign to show that what is on the left side has the same value as what is on the right side.

In the context of the problem, we set up the equation \(2x + 6 = x - 3\). This equation tells us that twice a number plus six is equal to three less than the number. The challenge is to manipulate this equation to find the value of \(x\).

To solve equations, one typical step is to isolate the variable on one side. This could involve adding or subtracting terms from both sides, or using multiplication or division. Equations not only help us solve problems involving unknown quantities but also enable us to describe generalized relationships and predict future outcomes in various fields.