Problem 31 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation or an inequality. Let x represent the number. The quotient of 35 and a number is 7.

Step-by-Step Solution

Answer

The equation derived from the sentence is \(\frac{35}{x} = 7\) and solving for x gives us \(x = 5\).

Step 1: Understand the relationship

First, figure out the relationship described in the sentence. The phrase 'quotient of 35 and a number' means we are dividing 35 by some number, which we'll represent with 'x'.

Step 2: Translate sentence to equation

Based on the relationship established, the sentence 'The quotient of 35 and a number is 7' can be translated into the equation \(\frac{35}{x} = 7\). This equation represents exactly the relationship described in the sentence, where x is the number we are looking for.

Step 3: Solve the equation for x

To solve for 'x', we can multiply both sides of the equation with 'x' and then divide by 7. So this gives us: \(x = \frac{35}{7}\).

Key Concepts

Inequalities

Inequalities are a way to express the relationship between two expressions that are not equal. They show how one expression can be greater or less than another. Unlike equations, which have a precise solution, inequalities describe a range of possible solutions.
Inequalities use symbols such as:

• '<' for less than.
• '>' for greater than.
• '≤' for less than or equal to.
• '≥' for greater than or equal to.

These symbols help us communicate the extent to which one side is larger or smaller than the other.
For instance, if we have an inequality like \( x > 5 \), it means that the solution includes all numbers greater than 5.
When writing an inequality, it’s essential to determine what the inequality should express. If the original statement describes a situation with a strict limit, like a maximum or a minimum, it's crucial to use the corresponding inequality sign.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They represent a particular value for each variable. The goal is often to simplify these expressions or use them in equations or inequalities to find values that satisfy specific conditions.
Variables, like 'x', are symbols used to represent unknown numbers. In the expression \( 3x + 5 \), '3' is the coefficient of 'x', and '5' is the constant term. Coefficients tell us how much the variable is multiplied by, while constants are fixed values that do not change in the expression.
To work with algebraic expressions:

• Combine like terms, which are terms that have the same variable raised to the same power.
• Use operations to simplify or expand the expression.
• Substitute known values for variables to calculate exact numbers.

Understanding how to manipulate algebraic expressions is foundational for solving equations and inequalities as they frequently appear in these tasks.

Solving Equations

Solving equations is one of the key processes in algebra. It involves finding the value of the variable that makes the equation true. An equation is like a balance scale, with each side representing equivalent values.
To solve an equation:

• Identify what operation is being applied to the variable.
• Perform inverse operations to isolate the variable. Inverse operations are the opposite actions, like using addition to undo subtraction or division to undo multiplication.
• Keep the equation balanced by performing the same operation on both sides of the equation.

In the solution provided:- The equation \( \frac{35}{x} = 7 \) represents a division.- To isolate 'x', multiply both sides by 'x' and then divide by '7'.- You find that \( x = 5 \).
This straightforward process shows how solving equations allows us to uncover the value of unknown numbers and understand their relationships in a given context.