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Chapter 2: Problem 3

Decide whether 6 is a solution of \(3(x+4)=5 x\) by substituting 6 for \(x .\) If is not a solution, explain why.

Short Answer

Expert verified
Yes, 6 is a solution because both sides of the equation simplify to 30.

Step by step solution

01

Identify the Given Equation

The given equation is: 3(x+4) = 5x
02

Substitute Value

Substitute 6 for x in the given equation 3(6+4) = 5(6)
03

Simplify Both Sides

On the left side: 3(10) = 30 On the right side: 5(6) = 30
04

Compare Both Sides

Both sides of the equation simplify to 30. The equation holds true when x = 6.
05

Conclusion

Since both sides are equal, 6 is a solution to the equation 3(x+4)=5x.

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Key Concepts

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

Substitution Method
To determine if a number is a solution to an equation, we use the substitution method. This involves replacing the variable in the equation with the given number. In our exercise, we needed to check if 6 is a solution for the equation \(3(x+4)=5x\). That meant substituting 6 in place of \(x\) in the equation. This approach is straightforward and helps verify whether the left-hand side and right-hand side of the equation are equal when a specific number is substituted.
Balancing Equations
Balancing equations is a crucial aspect of solving linear equations. The idea is to ensure that both sides of the equation represent the same value when simplified. In our example, once we substituted 6 into \(3(x+4)=5x\), we simplified both sides to check for equality. By simplifying \(3(10)\) to 30 and \(5(6)\) also to 30, we confirmed that both sides were balanced. When both sides of an equation yield the same value, it means the equation is balanced, indicating that our substitution was correct.
Simplifying Expressions
Simplifying expressions is a process that makes equations easier to solve and understand. After substituting a number for the variable, we often need to perform operations to simplify both sides of the equation. In our example, after substituting 6 for \(x\), we simplified the left side from \(3(6+4)\) to \(3(10)\), which then simplified to 30. Similarly, the right side \(5(6)\) simplified directly to 30. When simplifying, always perform arithmetic operations such as addition, multiplication, and combining like terms to convert the complex equation into simpler terms.
Verification of Solutions
Verification is the final step in solving equations. It ensures the solution satisfies the original equation. After simplifying both sides of the substituted equation \(3(x+4)=5x\), we compared the results. Both sides equaled 30. This confirms that the original equation holds true. Thus, 6 is indeed a solution. It's crucial to verify the solution, as it confirms the accuracy of all the steps in solving the equation. This process adds confidence in the correctness of our solution.

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