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Chapter 2: Problem 52

Solve equation, and check your solution. \(-\frac{3}{4} z=-21\)

Short Answer

Expert verified
z = 28

Step by step solution

01

Isolate the variable

To isolate the variable, multiply both sides of the equation by the reciprocal of \(-\frac{3}{4}\). The reciprocal of \(-\frac{3}{4}\) is \(-\frac{4}{3}\).
02

Perform the multiplication

Multiply both sides by \(-\frac{4}{3}\):\ \(-\frac{4}{3}\) \times \(-\frac{3}{4}z\) = \(-\frac{4}{3}\) \times (-21)\ This simplifies to: z = 28.
03

Check the solution

Substitute the value of z back into the original equation to check: \(-\frac{3}{4}\) \times 28 = -21\ Simplifying: \-21 = -21.

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Key Concepts

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

Multiplying by Reciprocals
When solving linear equations, you'll often need to get rid of fractions to isolate the variable. One powerful method is to multiply by the reciprocal. A reciprocal is simply the flipped version of a fraction. For example, the reciprocal of \(-\frac{3}{4}\) is \(-\frac{4}{3}\).

When you multiply a fraction by its reciprocal, the result is always 1. This is very useful in equations because it simplifies the equation. Let's consider the given problem: \(-\frac{3}{4} z = -21\). To remove the fraction, you multiply both sides by \(-\frac{4}{3}\). This helps in isolating the variable, which brings us to our next concept.
Isolating the Variable
The ultimate goal of solving an equation is to find the value of the unknown variable. To achieve this, the variable should be isolated on one side of the equation. Taking our example, after identifying the reciprocal \(-\frac{4}{3}\), we multiply both sides of the equation by it: \(-\frac{4}{3} \times -\frac{3}{4}z = -\frac{4}{3} \times -21\).

The left side simplifies to \(z\), because \(-\frac{4}{3} \times -\frac{3}{4} = 1\). On the right side, you perform another multiplication: \(-\frac{4}{3} \times -21 = 28\). Thus, you get \(z = 28\). Now that we have isolated the variable, the next task is to confirm the solution.
Checking Solutions
After isolating the variable and finding its value, it is crucial to check your solution to ensure it's correct. To do this, substitute the value back into the original equation and verify both sides are equal. For our example, take the solution \(z = 28\) and substitute it back into the original equation \(-\frac{3}{4} z = -21\).

Substituting gives: \(-\frac{3}{4} \times 28\). Simplifying this, you get \(-21\). The left side of the equation now equals the right side: \(-21 = -21\). This confirms that the solution \(z = 28\) is correct. Always check your solutions to avoid mistakes and ensure accuracy in your calculations.

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