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Chapter 11: Problem 5

Solve using the multiplication principle. Don't forget to check! $$ 84=7 x $$

Short Answer

Expert verified
The solution is \( x = 12 \).

Step by step solution

01

- Isolate the Variable

To solve for the variable \(x\), isolate it on one side of the equation. The equation given is \(84 = 7x\). Divide both sides of the equation by 7 to isolate \(x\). This gives: \[ \frac{84}{7} = \frac{7x}{7} \]which simplifies to: \[ 12 = x \]
02

- Check the Solution

To verify the solution, substitute \(x = 12\) back into the original equation. The original equation is \(84 = 7x\). Substitute \(x\) with 12, which yields: \[ 84 = 7 \times 12\]Calculate the right-hand side to confirm: \[ 84 = 84 \]Since both sides of the equation are equal, the solution is correct.

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Key Concepts

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

Multiplication principle
The multiplication principle is a crucial concept in solving linear equations. It states that you can multiply both sides of an equation by the same non-zero number without changing the equality. This principle is often used to eliminate fractions or decouple variables.
Here鈥檚 how it works in practice:
  • If you have an equation like \(a = b\), multiplying both sides by a non-zero number, say \(c\), gives \(ac = bc\).
  • This operation keeps the equation balanced and doesn't change the relationship between the variables or numbers involved.

In the exercise provided, the multiplication principle is used in reverse to divide both sides of an equation. By dividing both sides of \(84 = 7x\) by 7, you effectively isolated the variable, which we will discuss next.
Isolating variables
Isolating the variable is a key step in solving any equation. The goal is to rearrange the equation so that the variable is on one side of the equation by itself. This will show you the value of the variable.
Here鈥檚 a step-by-step breakdown for our example:
  • Start with the given equation: \(84 = 7x\).
  • To isolate \(x\), divide both sides by 7: \[ \frac{84}{7} = \frac{7x}{7} \].
  • This simplifies to: \[ 12 = x \], meaning that \(x = 12\).

This technique ensures that you鈥檝e correctly rearranged the equation without modifying the balance between the two sides.
Checking solutions
After isolating the variable and finding a solution, it鈥檚 important to check that your answer is correct. Verifying your solution ensures that you didn't make any arithmetic errors and that the solution satisfies the original equation.
Here鈥檚 how to check your solution:
  • Start with the original equation: \(84 = 7x\).
  • Substitute the found value of \(x\), which is 12, back into the equation: \[ 84 = 7 \times 12 \].
  • Calculate the right-hand side to see if it equals 84: \[ 84 = 84 \].

Since both sides of the equation are equal, the solution \(x = 12\) is indeed correct. This final step is crucial and shouldn't be overlooked, as it confirms the validity of your work.

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