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Chapter 5: Problem 43

Nine times a number is equal to 36 more than the number. Find the number.

Short Answer

Expert verified
The number is 4.5.

Step by step solution

01

Set Up the Equation

Let the unknown number be represented by the variable Let the unknown number be represented by the variable x. According to the problem, nine times the number (9x) is equal to 36 more than the number (x + 36). Set up the equation: 9x = x + 36.
02

Simplify the Equation

To solve for x, first get all terms involving x on one side of the equation. Subtract x from both sides: 9x - x = x + 36 - x, which simplifies to 8x = 36
03

Isolate the Variable

To solve for x, divide both sides of the equation by 8: 8x/8 = 36/8, which simplifies to x = 4.5

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are a fundamental part of solving equations. They consist of numbers, variables, and operators combined to represent a value. For example, in our problem, 'nine times a number' is represented by the algebraic expression \(9x\). Simultaneously, '36 more than the number' translates to \(x + 36\). These expressions help us model real-world situations in mathematical terms, making it easier to solve problems.
  • Algebraic expressions can include constants (like 36), variables (like \(x\)), and coefficients (like 9).
  • Using algebraic expressions allows us to set up equations that can be manipulated to find unknown values.
  • Understanding how to translate words into algebraic expressions is key in solving word problems.
Isolate Variable
Isolating the variable is crucial when solving linear equations. The goal is to get the variable (like \(x\)) by itself on one side of the equation, so we can determine its value. Let鈥檚 take a closer look at how we isolate the variable in our problem:
  • We started with \(9x = x + 36\).
  • Next, we moved all terms involving \(x\) to one side by subtracting \(x\) from both sides, giving us \(8x = 36\).
  • This step is essential to simplify the equation and focus solely on finding \(x\).
Once we have all variable terms on one side, we can easily solve for the variable. Dividing both sides by the coefficient of \(x\), in this case, 8, we get \(x = 4.5\). By isolating the variable, we make the equation simpler to manage and solve.
Simplify Equations
Simplifying equations helps us solve for unknown variables efficiently. Simplification involves combining like terms and performing operations step-by-step. Here鈥檚 what we did in our example:
  • The original equation was \(9x = x + 36\).
  • We simplified by subtracting \(x\) from both sides, resulting in \(8x = 36\).
  • Finally, we divided by 8 to isolate and solve for \(x\), giving us \(x = 4.5\).
By breaking down equations in this manner, we can make complex problems more manageable. Always remember to:
  • Combine like terms to reduce the equation's complexity.
  • Perform arithmetic operations step-by-step.
  • Ensure that each step logically follows from the previous one.
Simplifying equations is a fundamental skill in algebra, helping you to solve problems more quickly and accurately.

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