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Chapter 5: Problem 23

Solve the equations. $$19.43=-6.7 n$$

Short Answer

Expert verified
n = -2.9

Step by step solution

01

- Isolate the variable

To solve for the variable, isolate it on one side of the equation. Since the equation is \(19.43 = -6.7 n\), divide both sides by \(-6.7\): \(\frac{19.43}{-6.7} = n\).
02

- Perform the division

Next, perform the division on the left-hand side: \(19.43 \div -6.7\) which gives: \(n = -2.9\).

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Key Concepts

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

Isolate the variable
When solving a linear equation, the main goal is to find the value of the variable. To do this, we need to get the variable by itself on one side of the equation. This process is called 'isolating the variable.' In our example, the equation is given as:

\[19.43 = -6.7n\]

To start isolating \(n\), we need to move everything else to the other side of the equation. Here, we do that by dividing both sides by \(-6.7\):

\[\frac{19.43}{-6.7} = n\]

Notice that when dividing by a negative number, our variable will also be negative if the other side is positive. This initial step is crucial because having the variable alone simplifies the equation, making it easier to solve.
Division in equations
Division is a common operation used in isolating variables. In our specific case, we divided both sides of the equation by \(-6.7\) to isolate \(n\). This operation transforms the equation from:

\[19.43 = -6.7n\]

to:

\[n = \frac{19.43}{-6.7}\]

Here's a key tip: Whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced. It's like a see-saw; if you add or take from one side, you must do the same on the other side to keep it level.

Now, performing the division:

\[19.43 \text{ divided by } -6.7 = -2.9\]

So, we get:

\[n = -2.9\]

Division helps us simplify the equation by breaking down complex expressions into something more manageable.
Negative coefficients
A coefficient is a number that multiplies a variable. In our example, \(-6.7\) is the coefficient of \(n\) in the equation:

\[19.43 = -6.7n\]

Negative coefficients can sometimes be tricky. When dealing with negative coefficients, remember these key points:
  • Dividing by a negative number will reverse the sign of the variable.
  • Always perform the same operation on both sides to keep the equation balanced.

In our scenario, the negative coefficient \(-6.7\) means we have to be cautious about signs. When we divide both sides by \(-6.7\), we notice the variable's sign changes, resulting in:

\[n = \frac{19.43}{-6.7} = -2.9\]

Handling negative coefficients correctly ensures you don't make sign errors, which are common mistakes in solving linear equations.

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