Problem 70 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. The product of 6 and a number, increased by \(3,\) is 33

Step-by-Step Solution

Answer

The equation that corresponds to the given sentence is \(6x + 3 = 33\).

Step 1: Translate the sentence

The sentence, 'The product of 6 and a number, increased by 3, is 33' should be translated into a mathematical form. The words 'product of 6 and a number' indicate multiplication, represented by \(6x\). The phrase 'increased by 3' translates to '+ 3'. Lastly, 'is' signifies equality, represented by '='.

Step 2: Form the equation

Following from Step 1, we can form the sentence into an equation. It yields \(6x + 3 = 33\).

Step 3: Verify the equation

Check if the equation represents the sentence accurately. If it does, we have our correct mathematical representation of the sentence.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is like learning a new language, a language of numbers and symbols. Algebraic expressions are combinations of variables, numbers, and operations that represent a particular relationship or quantity without using an equals sign. For instance, when we talk about the 'product of 6 and a number', we think in terms of multiplication. In algebraic terms, this would be represented as '6x', where 'x' is a placeholder for the number in question.

An algebraic expression by itself is not a complete thought until it becomes part of an equation or an inequality. It’s the building block that helps us capture the relationships described in word problems. As such, when we see a phrase like 'increased by 3', it prompts us to add 3 to our existing expression, giving us '6x + 3'. Visualization can help: imagine you have six boxes, each containing 'x' items, and then you add another three items. This mental image translates directly into the algebraic expression.

Equations in Algebra

An equation in algebra is essentially a statement that two things are equal. It serves as a balance scale: what you do to one side, you must do to the other to maintain the balance. The equation '6x + 3 = 33' tells us that when we multiply a number 'x' by six and add three, the result is 33. It’s the mathematical way of expressing a balance between two expressions.

In solving equations, we utilize various operations to isolate the variable and find its value. This could involve subtraction, division, multiplication, or addition, often represented as inverse functions. It is critical to perform the same operation on both sides of the equation to keep it balanced. For example, to find the value of 'x' in the equation above, you would subtract 3 from both sides and then divide by 6, symbolizing both the logical and systematic approach in algebra.

Interpreting Mathematical Phrases

The tricky part for many students is interpreting mathematical phrases and translating them into equations. It's essential to recognize key terms that indicate specific operations: terms like 'product of' suggest multiplication, while 'sum' refers to addition. Conversely, 'difference' implies subtraction, and 'quotient' hints at division.

When a phrase includes words like 'increased by' or 'decreased by', they indicate addition or subtraction respectively. The word 'is' is typically the equivalent of the equal sign in an equation. By familiarizing oneself with these terms, students can systematically break down word problems into manageable parts.

For example, the phrase from the exercise 'the product of 6 and a number, increased by 3, is 33' includes the multiplication indicator 'product of', the addition indicator 'increased by', and the equal sign indicator 'is'. Recognizing these terms and translating them into mathematical statements is a critical skill in algebra that helps students to convert real-world problems into solvable equations.