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Chapter 1: Problem 49

Solve and check. \(17 \cdot n=119\)

Short Answer

Expert verified
Answer: n = 7

Step by step solution

01

Divide both sides by 17

To isolate n, we need to divide both sides of the equation by 17: \(\frac{17 \cdot n}{17}=\frac{119}{17}\)
02

Simplify the equation

Now, simplify the equation and find the value of n: \(n=\frac{119}{17}\) Thus, \(n=7\)
03

Check the solution

In order to confirm our solution is correct, substitute the found value of n back into the original equation: \(17 \cdot n=119\) \(17 \cdot 7=119\) \(119=119\) Since both sides of the equation are equal, we can confirm that our solution of n=7 is correct.

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Key Concepts

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

Solving Equations
When you solve a linear equation, your goal is to find the value of the variable that makes the equation true. In this exercise, we deal with the equation \(17 \cdot n = 119\). This is a simple linear equation with one variable, \(n\). To solve for \(n\), we need to perform operations that will leave \(n\) alone on one side of the equation. This often means isolating \(n\) by using opposite operations to "undo" what has been done to it. The process usually involves:
  • Identifying the operation affecting the variable (in this case, multiplication by 17).
  • Using the inverse operation (here, division) to isolate the variable.

By applying these steps systematically, you can solve many linear equations with ease.
Division
Division is a key arithmetic operation that helps in solving equations, especially when you have multiplication involved, like in our equation \(17 \cdot n = 119\). Division is the inverse of multiplication. To isolate \(n\), we divide both sides of the equation by 17, as mentioned in the original solution.Here's how division works in the context of an equation:
  • Ensure that you perform the division on both sides of the equation to maintain balance and equality.
  • Simplify the result after dividing, which will directly give you the value of the variable.
After dividing, we got \(n = \frac{119}{17}\), which simplifies to \(n = 7\). This shows that the proper application of division can efficiently help solve for unknowns in your equations.
Verification
Verification is an essential final step in solving equations. It ensures that the solution obtained is indeed correct. For the equation \(17 \cdot n = 119\), after finding that \(n = 7\), we substitute this value back into the original equation to confirm its validity.Here鈥檚 how to verify:
  • Replace the variable in the original equation with the solution found.
  • Perform the necessary arithmetic operations.
  • Check if both sides of the equation are equal.
When you substitute \(n = 7\) back, it results in \(17 \cdot 7 = 119\), confirming that \(119 = 119\). Since the left and right sides match, the solution is verified as correct. Verification helps prevent errors and gives confidence in your calculated solution.

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