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Chapter 2: Problem 37

Determine whether the given number is a solution of the equation. $$x+3=6 ; 3$$

Short Answer

Expert verified
Yes, the number 3 is a solution to the equation \(x + 3 = 6\).

Step by step solution

01

Understand the Problem

The exercise provides an equation \(x + 3 = 6\) and proposes a solution, which is the number 3. The objective is to validate if the number 3 is indeed the solution of the given equation.
02

Substitute the Proposed Solution

To determine whether 3 is a solution to the equation, substitute 3 in place of x in the equation. The equation will become \(3 + 3 = 6\).
03

Simplify and Check

Simplify the left side of the equation. \(3 + 3\) simplifies to 6. Because both the left side (6) and the right side (6) of equation are equal after the substitution, it is proven that the number 3 is indeed a solution to the equation \(x + 3 = 6\).

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Key Concepts

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

Solution Verification
Solution verification is the process of checking if a proposed solution for an equation is correct. In this context, an equation like \(x + 3 = 6\) requires substituting the solution candidate into the equation to see if it holds true. Solution verification involves critical steps that ensure accuracy and demonstrate a clear understanding of the problem.
  • Proposed Solution: Start by identifying the solution you need to verify. In our example, the number 3 is proposed as the potential solution.
  • Substitution: Replace the variable with the proposed solution in the equation. Here, substitute \(x\) with 3.
  • Comparison: Simplify the equation with your substitution and compare both sides. If they match, the solution is correct; otherwise, it is not.
Bringing all these steps together verifies if the number satisfies the original equation, confirming its validity.
Basic Algebra
Basic algebra involves operations with numbers and variables, setting the foundation for solving equations. In linear equations like \(x + 3 = 6\), you are asked to find the value of \(x\) that makes both sides balanced. Here's how basic algebra principles apply to this problem:
  • Understanding Variables: A variable, like \(x\), represents an unknown number. Your job is to find its value.
  • Equation Balancing: Ensure that whatever operations you perform, both sides of the equation remain equal.
  • Adding & Subtracting: Use basic operation techniques, like addition or subtraction, to isolate the variable and solve for it. In our equation, think of subtracting 3 from both sides to solve for \(x\).
Mastering basic algebra is crucial as it forms the basis for tackling more complex equations and mathematical situations.
Substitution Method
The substitution method is a fundamental technique used in algebra to solve equations or verify solutions. This approach involves substituting a known value into an equation, allowing you to check if the equation holds true or solve for another variable. Here鈥檚 how it works in our example:
  • Replacement: Substitute the given solution, 3, into the equation for the variable \(x\).
  • Simplification: After substitution, simplify both sides of the equation鈥攔esulting in \(3 + 3 = 6\).
  • Verification: If both sides are equal after the substitution, as they are here, the solution is verified.
Using the substitution method helps confirm potential solutions and is a powerful tool for solving equations by rendering them easier to manage and solve. It systems control and clarity, making it central to mathematical problem-solving strategies.

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Most popular questions from this chapter

Solve. After a 2 -for- 1 stock split, each shareholder owned twice as many shares as before. If 2,570 shares are owned after the split, how many were owned before?

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