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Chapter 1: Problem 57

Determine whether the given number is a solution of the equation. $$2(w+1)=3(w-1) ; 7$$

Short Answer

Expert verified
No, the number 7 is not a solution to the equation.

Step by step solution

01

Problem Identification

The problem presents an equation 2(w + 1) = 3(w - 1), and a number, 7 which needs to be checked as a solution to the equation.
02

Substitution of number into equation

Substitute 7 for w in the equation, we get 2(7 + 1) = 3(7 - 1).
03

Simplify the equation

Solve it step by step, first compute the expressions inside the brackets, we get 2(8) = 3(6), which simplifies to 16 = 18.
04

Verify the Solution

The resulting left side of the equation (16) is not equal to the right side (18), so the number 7 doesn't solve the equation.

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Key Concepts

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

Solution Verification
Verifying a solution to an algebraic equation is an essential step in ensuring correctness. The process involves substituting the given value, in this case, the number 7, into the equation to check if it leads to a true statement.
Here's how it works:
  • Substitute the number where the variable appears in the equation.
  • Calculate both sides separately.
  • Compare the two results.
For the equation \( 2(w+1) = 3(w-1) \), upon substituting \( w = 7 \), it simplifies to an untrue equation \( 16 = 18 \).
Thus, 7 is not a solution. This simple check highlights discrepancies and ensures you're working with the correct values.
Substitution Method
The substitution method is a reliable technique for testing potential solutions in algebra.
When you have an equation and a proposed solution, such as the number 7 in \( 2(w+1) = 3(w-1) \), substitution allows you to verify if it satisfies the equation.
Steps to use substitution:
  • Identify the variable in the equation鈥攈ere it's \( w \).
  • Replace this variable with the given number, which is 7.
  • Solve the new equation to see if both sides equal.
In our example, substituting \( w = 7 \) transforms the equation to \( 2(8) = 3(6) \) leading to \( 16 = 18 \).
Since the equality doesn't hold, the substitution method shows 7 isn鈥檛 a valid solution. This method is simple but powerful for verifying solutions in algebra.
Simplifying Expressions
Simplifying expressions is a foundational skill in algebra that involves reducing expressions to their simplest form.
When you substitute variables, simplifying helps you accurately verify the equation.
In the given problem, after substituting \( w = 7 \) into \( 2(w + 1) = 3(w - 1) \), the expression becomes \( 2(8) = 3(6) \).
  • Calculate inside the parentheses: \( 7 + 1 = 8 \) and \( 7 - 1 = 6 \).
  • Multiply: \( 2 \times 8 = 16 \) and \( 3 \times 6 = 18 \).
By simplifying each side, you uncover whether the equation holds true. Since \( 16 eq 18 \), the values clearly do not match, confirming the number isn't a solution.
Simplifying reveals clear results and avoids errors when dealing with complex equations.

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Most popular questions from this chapter

Simplify each algebraic expression. $$2 x-10 x$$

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Write a problem that can be solved by finding the difference between two numbers. At least one of the numbers should be negative. Then explain how to solve the problem.

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