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Chapter 3: Problem 14

Determine if the given value is a solution to the equation. $$-2 w+9+w-11=-1 ; w=-3$$

Short Answer

Expert verified
After substituting the given value of \(w = -3\) into the equation \(-2 w + 9 + w - 11 = -1\) and performing the arithmetic operations, we find that both sides of the equation are not equal, i.e., \(-2 鈮� -1\). Therefore, the given value of \(w = -3\) is not a solution to the equation.

Step by step solution

01

Substitute the given value of w in the equation

Replace \(w\) with \(-3\) in the equation \(-2w + 9 + w - 11 = -1\). \[ -2(-3) + 9 + (-3) - 11 = -1 \]
02

Simplify the equation

Perform the arithmetic operations in the equation. \[ (6) + 9 - 3 - 11 = -1 \]
03

Further simplify the equation

Continue performing the arithmetic operations. \[ 12 - 14 = -1 \]
04

Isolate the variable

Check if both sides of the equation are equal. \[ -2 = -1 \] Since \(-2 鈮� -1\), the given value of \(w = -3\) is not a solution to the equation \(-2 w + 9 + w - 11 = -1\).

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Key Concepts

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

Equation Solving
Solving an equation involves finding the value of a variable that makes the equation true. In simpler terms, it鈥檚 like a balancing act where both sides of the equation must have the same value.
Each side can contain numbers, variables (like "w"), and arithmetic operations (addition, subtraction, etc.). The goal is to determine the specific value that fits seamlessly into this balancing act.
In the given exercise, we are tasked with checking if the value \(w = -3\) satisfies the equation \(-2w + 9 + w - 11 = -1\).
Let's delve further into the methods used to determine if this specific value is a solution.
Substitution Method
The substitution method is a technique to check whether a given value satisfies an equation. Essentially, it involves replacing the variable in the equation with a provided number.
Let's break it down with our equation: Start by substituting \(w = -3\) into the equation \(-2w + 9 + w - 11 = -1\).
This transforms the equation as follows: \(-2(-3) + 9 + (-3) - 11 = -1\).
  • "-2(-3)" involves replacing \("w "\) with \(-3\).
  • The rest of the equation remains the same until further arithmetic operations are done.
This substitution allows us to verify if the calculated expressions equal the original right-hand side value, \(-1\).
It's a straightforward plug-and-check process!
Arithmetic Operations
Arithmetic operations are used extensively in simplifying and solving equations. They include addition, subtraction, multiplication, and division.
Once the substitution of \(w = -3\) is done, the equation \(-2(-3) + 9 + (-3) - 11 = -1\) needs to be simplified using arithmetic operations:
  • First, calculate \(-2(-3)\) which results in \(6\). This is because multiplying two negative numbers yield a positive result.
  • Next, add \(9\) to \(6\), which gives \(15\).
  • Subtract \(3\) from \(15\), leading to \(12\).
  • Then, subtract \(11\) from \(12\). This results in \(1\).
The final step reveals that the simplified left side is \(1\), not \(-1\), leading to the conclusion that our solution is incorrect.
Checking Solutions
Checking solutions involves determining whether the value we have substituted satisfies the equation.
After conducting the arithmetic operations, we noted that the left side of the equation equals \(1\) instead of \(-1\).
This means that \(-2 = -1\) is **not** a true statement, and therefore, the given value \(w = -3\) does not satisfy the equation \(-2w + 9 + w - 11 = -1\).
It's crucial to always confirm that both sides of the equation are equivalent after substitution and simplification.
This step essentially confirms or denies the accuracy of our proposed solution, and ensures we either find the correct value or understand why a substitution doesn't work.

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