Problem 48 - Algebra 1

Question

Translate the sentence into an equation or an inequality. -9 is equal to a number \(y\) decreased by \(21 .\)

Step-by-Step Solution

Answer

\(y - 21 = -9\)

Step 1: Analysis of Sentence

The first part of the sentence has '-9' which is the result. The last part of the sentence has 'a number y decreased by 21'. 'Decreased by' implies subtraction in the mathematical operations. We need to form a equation using these items.

Step 2: Formulate Equation from Sentence

The sentence is translated as: the result of subtracting 21 from a certain number y is equal to -9. In mathematical terms, this can be expressed as \(y - 21 = -9\). The minus sign in front of the 9 is because of 'decreased by', which indicates subtraction.

Step 3: Check the Equation

Check the equation to ensure that it matches the original sentence. The left part of the equation should mirror 'a number y decreased by 21' and the right part should correspond to '-9'. Verify whether \(y - 21 = -9\) accurately reflects '-9 is equal to a number y decreased by 21'

Key Concepts

Subtraction

Subtraction is one of the fundamental operations in mathematics. It involves taking away one quantity from another. When you see terms like 'decreased by', 'less', or 'minus', these usually indicate subtraction is happening. In an equation, subtraction is represented by the minus sign

• 'Decreased by' means you subtract a certain amount from the original.
• For example, if you have 10 and you decrease it by 4, you would write it as \(10 - 4\).

In the context of our exercise, 'a number \(y\) decreased by 21' translates to subtracting 21 from \(y\): \(y - 21\)

Translation of Sentences into Equations

Translating sentences into mathematical equations is a crucial skill in algebra. It enables you to convert real-world situations into a mathematical format that can be solved.

• Begin by identifying keywords in the sentence that suggest mathematical operations.
• Translate each section of the sentence into a mathematical expression.
• Use symbols like \(+\), \(-\), \(=\) to form the complete equation.

In the provided exercise, 'is equal to' suggests an equation. Therefore, \(-9\) is set to equal 'a number \(y\) decreased by 21', resulting in \(y - 21 = -9\). This process transforms a descriptive sentence into a precise mathematical statement.

Variable Identification

Variables are symbols used to represent unknown values or values that can change. In mathematics, they are often denoted by letters such as \(x\), \(y\), or \(z\). Identifying variables is the first step in solving an equation that involves unknowns.

• Look for phrases like 'a number' or 'an amount' in the sentence, which often imply variables.
• Assign a letter, such as \(y\), to represent this unknown quantity.

In the given exercise, the phrase 'a number \(y\)' identifies \(y\) as the variable we need to solve for. It's crucial to clearly define and consistently use variables within your equations.