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Chapter 1: Problem 2

Which of the numbers following the given equation are solutions of the given equation? $$ \frac{x}{7}=6 ; 49,42,43,45 $$

Short Answer

Expert verified
42 is the solution to the equation.

Step by step solution

01

Identify the Equation

The given equation is \(\frac{x}{7} = 6\). We need to find which values of \(x\) satisfy this equation from the list provided: 49, 42, 43, 45.
02

Solve the Equation for \(x\)

Multiply both sides of the equation \(\frac{x}{7} = 6\) by 7 to solve for \(x\):$$x = 6 \times 7.$$Calculating gives:$$x = 42.$$
03

Verify the Solution

Check the list of numbers: 49, 42, 43, 45, to see which one equals the solution \(x = 42\). The number 42 is in the list.
04

Conclusion

Since 42 is in the list and satisfies the equation \(\frac{x}{7} = 6\) when we plug it back in, it is a solution to the equation.

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Key Concepts

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

Substitution Method
The substitution method is a powerful tool used to solve equations, especially when you want to check if a proposed solution satisfies a given equation.
To apply this method, follow these straightforward steps:
  • Start with the equation you are given. In our example, it's \( \frac{x}{7} = 6 \).
  • Substitute each value you want to check into the equation, replacing the variable with the number. For instance, trying 49, replace \( x \) with 49 to get \( \frac{49}{7} = 6 \), which tests if it makes the equation true.
  • If after substitution both sides of the equation are equal, then that number is a solution.
  • If they are not equal, then try the next number until all options have been tested.
In our example, substituting 42 for \( x \) results in \( \frac{42}{7} = 6 \), confirming that 42 is indeed a solution.
Verifying Solutions
Verifying a solution is all about ensuring that the calculation has been done correctly, and the right value has been found. It's like double-checking your work to make sure everything is spot-on.

Here鈥檚 how you can verify a solution:
  • Once you have a potential solution from solving the equation, substitute it back into the original equation.
  • Recalculate to ensure both sides of the equation balance out.
  • If they do, the solution is verified as correct.
In our situation, we calculated the solution \( x = 42 \) in step 2. By plugging 42 back into \( \frac{x}{7} = 6 \), you get \( \frac{42}{7} = 6 \), proving that the solution is correct.
Multiplication Property of Equality
The multiplication property of equality is a fundamental concept useful in solving linear equations. It lets us manipulate equations to find the value of the unknown variable.
Here鈥檚 the concept explained:
  • When you have an equation, you can multiply both sides by the same non-zero number, and the equality will still hold.
  • This property is key when dealing with fractions or dividing terms because it helps to clear the fraction to isolate the variable.
  • In our equation \( \frac{x}{7} = 6 \), by multiplying both sides by 7, you remove the fraction, which simplifies to \( x = 42 \).
Using the multiplication property, we efficiently solve the equation, showing its practicality in arithmetic operations.

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Most popular questions from this chapter

Simplify the given expression. $$ 12 \cdot 5^{2}+8 \cdot 9+4 $$

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