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Chapter 1: Problem 65

Decide whether the given number is a solution of the equation. \(0.5(x-4)=80 ; \quad 20\)

Short Answer

Expert verified
20 is not a solution of the equation.

Step by step solution

01

Understand the problem

Determine if the number 20 is a solution for the equation \(0.5(x - 4) = 80\). This means substituting 20 for \(x\) in the equation and checking if both sides are equal.
02

Substitute the given number into the equation

Replace \(x\) with 20 in the equation. The equation becomes: \[0.5(20 - 4) = 80\]
03

Simplify inside the parentheses

Subtract 4 from 20: \[0.5(16) = 80\]
04

Perform the multiplication

Multiply 0.5 by 16: \[8 = 80\]
05

Compare both sides of the equation

Check if 8 is equal to 80. Since they are not equal, 20 is not a solution of the equation.

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Key Concepts

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

Solving Equations
Solving equations is one of the fundamental skills in algebra. Let's make it simple. An equation is like a balance scale. For the scale to be balanced, both sides must be equal. This means whatever you do to one side of the equation, you should do to the other side too.

Imagine you have an equation: \( 0.5(x - 4) = 80 \). If you want to know if a number is a solution, you must replace (or 'substitute') the variable with that number and check if both sides are still equal. In this case, we need to see if 20 is a solution. The step-by-step process is straightforward:

  • Substitute the number for the variable
  • Simplify the expression
  • Check if both sides of the equation match
Understanding these steps helps you find out if a number is a solution to an equation.
Substitution Method
The substitution method is used to solve equations by replacing variables with given values. Here, we're checking if 20 is a solution for the equation \( 0.5(x-4)=80 \). Here's how we do it:

Step-by-step:
  • Replace the variable \( x \) with 20. So, the equation \( 0.5(x-4) \) becomes \( 0.5(20-4) \).
  • Perform operations inside the parentheses first. Thus, we subtract 4 from 20, resulting in 16.
  • The equation now looks like this: \( 0.5 \times 16 = 80 \).
  • Next, multiply 0.5 by 16, which equals 8.
  • Finally, compare the result with 80. Since 8 is not equal to 80, 20 is not a solution.
This simple process helps you decide if a particular value satisfies the equation.
Verifying Solutions
Verifying solutions involves checking if a substituted value balances the equation. This can be essential in ensuring the correctness of a solution.

Let's break down the verification process:
  • First, substitute the variable with the given number. For instance, we replaced \( x \) with 20 in \( 0.5(x-4)=80 \).
  • Next, simplify the equation step by step. Subtract 4 from 20 to get 16, then calculate \( 0.5 \times 16 \), which is 8.
  • Then, compare the simplified result (8) with the original right-hand side of the equation (80).
  • If both sides match, the number is a solution. However, since 8 does not equal 80 in our example, 20 is not a solution.
Verification is your final check to ensure your solution is correct. It鈥檚 like double-checking your work to avoid mistakes.

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