Problem 63 - Introductory Algebra for College Students

Question

Solve and check: \(\frac{4}{5} x=-16\) (Section 2.2, Example 3)

Step-by-Step Solution

Answer

The solution to the equation \(\frac{4}{5} x=-16\) is \(x=-20\).

Step 1: Identifying the equation

The equation given is \(\frac{4}{5} x=-16\). Our goal is to solve for \(x\).

Step 2: Isolate the variable

To isolate \(x\), multiply both sides of the equation by the reciprocal of \(\frac{4}{5}\), which is \(\frac{5}{4}\). This gives us:\[x = \frac{5}{4} \times -16\]

Step 3: Calculate the right hand side

Multiplying \(\frac{5}{4}\) with -16 gives -20,\[x = -20.\]Thus, -20 is the solution to the given equation.

Step 4: Confirm the result

Substituting -20 into the original equation and solving, we have:\[\frac{4}{5} \times -20 = -16.\]This is a true statement, confirming that -20 is indeed the solution to the equation.

Key Concepts

Isolating the Variable

When solving linear equations, our main task is to find the value of the variable that satisfies the equation. In the given exercise, the equation is \(\frac{4}{5} x = -16\). The variable we want to solve for here is \(x\). To make this happen, we need to isolate \(x\), which means having it on one side of the equation all by itself.To do this, we need to get rid of the fraction that is multiplied by \(x\). Fractions can be tricky, but knowing how to handle them is crucial:

• The first step is to understand that the number \(\frac{4}{5}\) represents a multiplication with \(x\).
• To isolate \(x\), we need to "undo" this multiplication by doing the opposite operation, which is a key problem-solving strategy in algebra.

Reciprocal Multiplication

The "undo" operation for multiplication by a fraction is to multiply by its reciprocal. The reciprocal of a fraction \(\frac{a}{b}\) is simply \(\frac{b}{a}\). This key action allows us to eliminate the fraction and simplify the equation.For the equation \(\frac{4}{5} x = -16\), the reciprocal of \(\frac{4}{5}\) is \(\frac{5}{4}\). Multiply both sides of the equation by \(\frac{5}{4}\):

• This step is crucial because multiplying a number by its reciprocal gives 1, effectively cancelling out the fraction on one side.
• After doing this, our equation simplifies to \(x = \frac{5}{4} \times -16\).
• Perform the multiplication: \(\frac{5}{4} \times -16 = -20\).

Thus, \(x = -20\). Now \(x\) is isolated and we have our solution. Multiplying by the reciprocal is a powerful tool in solving equations involving fractions.

Checking the Solution

Once we think we have found the solution, it's very important to check if it actually works. This verification step ensures that our solution is correct and the equation holds true.To check the solution \(x = -20\) for the original equation \(\frac{4}{5} x = -16\):

• Substitute \(x\) with \(-20\) in the original equation.
• Calculate: \(\frac{4}{5} \times -20\).
• This must equal \(-16\), the right side of the original equation.

Compute this: \(\frac{4}{5} \times -20 = \frac{-80}{5} = -16\), which confirms our solution is correct.Checking ensures that no errors were made in the calculation, and it boosts confidence in the solution process. It's like a final "thumbs up" for the answer!