Question

Write the verbal sentence as an equation, or an inequality. A number \(t\) increased by the sum of seven and the square of another number \(s\) is ten.

Short answer

The equation that represents the given verbal sentence is \(t + 7 + s^2 = 10\).

Step-by-step solution

Identify the given elements

We have a number \(t\), and we have another number \(s\). There's an operation that involves 'increasing by', which translates into addition in mathematical context. Then, there is also the phrase 'sum of seven and the square of \(s\)', which indicates that we should add seven to the square of \(s\). Finally, it is given that the result is ten.

Translate the sentence into an equation

Now we need to translate the sentence into an equation, making sure to translate each part correctly. Thus, the sentence 'A number \(t\) increased by the sum of seven and the square of another number \(s\) is ten' can be translated into the equation as \(t + (7 + s^2) = 10\).

Simplify the equation if necessary

As a last step, it's crucial to check if the equation can be further simplified. In this instance, parentheses can be removed due to the absence of multiplication or division next to the parentheses. So, our final equation is \(t + 7 + s^2 = 10\).

Key concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental to grasping the language of algebra. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as +, -, *, /). The variables represent unknown values and are often used in equations to solve problems.

For instance, if we're told that the sum of seven and the square of another number s is part of our expression, we would write this as '7 + s²'. Here, 's²' represents the square of s, and the + sign indicates that seven is being added to that square. Algebraic expressions are the building blocks for writing equations, as they help us translate real-world scenarios into a language that can be manipulated mathematically to find solutions.

Writing Equations

Writing equations is a skill that involves translating verbal sentences into mathematical ones. This requires a strong understanding of the vocabulary used in algebra. Words like 'increased by', 'sum of', and 'is' have specific mathematical meanings. 'Increased by' suggests addition, 'sum of' indicates the result of addition, and 'is' usually means 'equals' in algebra.

As seen in the provided exercise, the verbal sentence is 'A number t increased by the sum of seven and the square of another number s is ten'. Key phrases to identify include 'a number t', 'increased by', 'sum of seven and the square of another number s', and 'is ten'. These translate to t, +, (7 + s²), and = 10 respectively. Thus, the complete equation we write is t + (7 + s²) = 10. Getting comfortable with this translation process is essential for moving from problems stated in words to a mathematical representation that can be analyzed and solved.

Problem-Solving in Algebra

Problem-solving in algebra involves several steps that start with understanding the problem and end with the solution. Initially, it's essential to identify the known and unknown quantities and understand how they are related. Next, translating the verbal description into a mathematical model—in this case, an equation—is crucial. After setting up the equation, the goal is to solve for the unknown variable.

In the exercise, after writing the equation t + (7 + s²) = 10, the next task is simplification. While the given problem does not involve solving for a particular variable, in many cases, simplifying can lead to easier manipulation and solution of the equation. Simplification might include combining like terms, eliminating parentheses, or isolating the variable. These algebraic techniques enable us to solve for the unknowns in the equations we've written, thus providing the solution to the problem initially described verbally.