Problem 8 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation or an inequality. The product of 7 and a number y is 42.

Step-by-Step Solution

Answer

The equation corresponding to the sentence is \(7*y = 42\).

Step 1: Identify the Variables

From the statement, we can identify the number '7' and a variable 'y' that we are not given a specific value for. 'y' is our unknown variable.

Step 2: Identify Operation and Relationship

The term 'product' implies multiplication. The statement says that the product of 7 and y equals 42. So, we use the mathematical symbol for multiplication, which is '*'. The term 'is' implies equality, hence we use the mathematical symbol for equality, which is '='.

Step 3: Write as an Equation

Now, we use the identified variables, operation, and relationship to write the expression as an equation. We get \(7*y = 42\). This equation properly conveys the statement given in the problem.

Key Concepts

Understanding Algebra

Algebra is like the language mathematicians use to describe relationships between numbers and variables. It involves using letters or symbols to represent unknown values or quantities. These unknowns are referred to as variables. Algebraic equations allow us to express problems, find unknown values, and show how different elements are connected. In this exercise, the variable is "y," an unknown number that, when multiplied by 7, equals 42. Understanding algebra starts with recognizing these variables and finding the operations involved, such as multiplication, which is key in this problem.
Algebra helps simplify complex problems and can be used across various fields like engineering, science, and economics. Once you understand how to set up algebraic expressions, solving equations becomes a straightforward process, helping you to make sense of the activities the symbols signify.

Using Multiplication

Multiplication is one of the four fundamental arithmetic operations. It involves combining equal groups. In algebra, multiplication is frequently used to connect a constant with a variable, showing how many times the constant is added to itself. In the exercise, the keyword 'product' indicates multiplication.
For example, in the expression 'the product of 7 and a number y,' we multiply the constant 7 with the variable y: this gives us the expression 7*y.

• Multiplication helps in describing situations where you have multiple groups of the same size.
• It is denoted by different symbols such as '*', 'x', or even simply placing numbers and variables next to each other (e.g., 7y).

Thus, understanding and identifying multiplication in problems is vital for setting up correct equations in algebra.

The Role of Variables

Variables are symbols used in algebra to represent unknown numbers or values. They are typically denoted by letters like x, y, or z. Variables allow us to construct equations that describe relationships in mathematical terms. In this exercise, the variable 'y' stands in for a number we do not yet know.
Using variables allows us to form general expressions or equations, which can then be solved to find particular numbers. This makes it possible to find solutions for a variety of problems without directly knowing every value involved initially.

• Variables can change value depending on the problem context.
• They serve as placeholders that enable us to perform operations and solve equations.

Understanding how to work with variables is crucial when solving equations, as it paves the way for finding unknown values and exploring their relationships with other numbers.