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Chapter 2: Problem 103

For exercises 103-106, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(3(x-2)-(x+5)=1\) Incorrect Answer: \(3(x-2)-(x+5)=1\) $$ \begin{aligned} 3 x-6-x+5 &=1 \\ 2 x-1 &=1 \\ \frac{+1}{2 x+0} &=2 \\ \frac{2 x}{2} &=\frac{2}{2} \\ x &=1 \end{aligned} $$

Short Answer

Expert verified
The mistake was in solving \(2x-1=1\); the correct solution is \ x=6 \.

Step by step solution

01

- Identify the Mistake

Read through the incorrect solution provided and identify where the mistake is made. The mistake in the solution is in the step where the equation \(2x-1=1\) is solved incorrectly. The equation \( \frac{+1}{2x+0}=2 \ \) is not a valid transformation.
02

- Rewrite the Original Problem

Start with the original equation \(3(x-2)-(x+5)=1 \): \[ 3(x-2)-(x+5)=1 \]
03

- Distribute the Constants

Apply the distributive property: \[ 3(x-2)-(x+5)=1 \] becomes\[ 3x-6-x-5=1 \]
04

- Combine Like Terms

Combine the terms involving \( x\):\[ 3x-x-6-5=1\] This simplifies to\[ 2x-11=1 \]
05

- Solve for the Variable

Add 11 to both sides to isolate the term with \( x \):\[ 2x - 11 + 11 = 1 + 11 \] This gives \[ 2x = 12 \]
06

- Final Step

Divide both sides by 2 to solve for \( x \):\[ x = \frac{12}{2} = 6 \]

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Key Concepts

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

Distributive Property
When solving linear equations like \(3(x-2)-(x+5)=1\), the distributive property is crucial. This property allows us to multiply a single term by each term inside a parenthesis. Here's how it works for our problem:
Start with: \[3(x-2)-(x+5)=1\]
Apply the distributive property: \[3(x) + 3(-2) - (x) - 5 = 1\]
Resulting in: \[3x - 6 - x - 5 = 1\]
As you can see, we distributed the 3 to both x and -2, and the -1 (since there is a negative sign before the parenthesis) to x and 5 respectively.
Combining Like Terms
After using the distributive property, we have the expression: \[3x -6 - x - 5 = 1\] The next step in solving equations is to combine like terms鈥攖erms with the same variable. Here's how it's done:
Identify the like terms: \[3x - x\] and \[-6 - 5\]
Combine the like terms: \[3x - x = 2x\] and \[-6 - 5 = -11\]
This results in: \[2x - 11 = 1\]
Combining like terms simplifies the equation, making it easier to solve for the variable.
Isolating Variables
Once we have a simplified equation like \[2x - 11 = 1\], our goal is to isolate the variable (x) on one side of the equation. This process involves a few steps:
Add 11 to both sides to remove the constant term from the left side: \[2x - 11 + 11 = 1 + 11\] This simplifies to: \[2x = 12\]
Divide both sides by 2 to solve for x: \[x = \frac{12}{2} = 6\]
By isolating the variable, we determine that \[x = 6\]
Remember, isolating variables involves performing inverse operations鈥攁ddition cancels out subtraction, and division cancels out multiplication.

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