Problem 87 - Algebra A Combined Function

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. One increased by two equals the quotient of nine and three.

Step-by-Step Solution

Answer

The equation is \( 1 + 2 = \frac{9}{3} \).

Step 1: Identify Known Quantities

The exercise mentions three different values: one, two, nine, and three. These are known quantities and will not involve any unknowns.

Step 2: Express Unknowns

The exercise does not mention any unknown numbers that need to be represented by a variable like \(x\). Hence, there will be no use of \(x\) in this equation as the exercise doesn't involve unknowns.

Step 3: Translate Sentence into Equation

The phrase "One increased by two" opens with the number 1 and then adds 2 to it, which can be written as \(1 + 2\). Next, "equals the quotient of nine and three" means the result is \(\frac{9}{3}\). Therefore, the complete equation is: \[ 1 + 2 = \frac{9}{3} \]

Step 4: Solve the Equation

Perform the calculations to verify:- Calculate the left side: \(1 + 2 = 3\).- Calculate the right side: \(\frac{9}{3} = 3\).Since both sides are equal, the equation holds true.

Key Concepts

Translating Words into Math Equations

Writing mathematical equations from word problems is like translating a language. In math, words like "increased by," "product of," or "equals" are key signals to formulating equations. To tackle translating words into math, look for:

• The sum, difference, product, or quotient – these operations often signal the arithmetic to perform.
• Equality and inequality phrases like "equals," "greater than," or "less than."
• Numbers and phrases that imply numbers, like "a dozen" for twelve.

In the given exercise, "One increased by two equals the quotient of nine and three," the word "increased" indicates an addition operation. "Equals" signals a balance, meaning both sides of our equation must be equal in value. Understanding these clues helps us translate the sentence into the math equation: \[ 1 + 2 = \frac{9}{3} \]

Solving Basic Equations

Once you have your equation, solving it involves performing arithmetic operations needed to simplify both sides. The goal is usually to isolate the variable or, when no variables are present, confirm both sides of the equation are equivalent.
Here's how you solve a simple equation step-by-step:

• Perform any addition, subtraction, multiplication, or division as indicated by the equation.
• Simplify both sides to ensure balance.
• Check that the final, simplified values are equal, proving the equation is correct.

In the example equation \(1 + 2 = \frac{9}{3}\), you would add on the left to get 3, and divide on the right to get 3. Because both sides are equal, the equation is solved successfully, confirming the relationship initial sentence described.

Basic Algebra Concepts

Algebra is the study of mathematical symbols and the rules for manipulating these symbols. In basic algebra, a variable like \(x\) is often used to represent an unknown number. But not every problem includes unknowns or requires a variable.
Here are core ideas in basic algebra:

• An equation is a mathematical statement indicating that two expressions are equal.
• An inequality compares two expressions to show a relationship that isn’t equality, using >, <, ≥, or ≤.
• Basic algebraic operations involve applying the arithmetic operations of addition, subtraction, multiplication, and division to the variables and constants.

In our exercise, the statement forms a basic equation rather than involving algebraic variables. This showcases how algebra isn't always about solving for unknowns but can simply be about verifying a true mathematical statement.