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Chapter 2: Problem 95

Solve each equation. $$ \frac{3}{4} x=21 $$

Short Answer

Expert verified
The solution is \(x = 28\).

Step by step solution

01

Understand the equation

The given equation is \(\frac{3}{4} x = 21\). The goal is to solve for \(x\).
02

Isolate the variable

To isolate \(x\), multiply both sides of the equation by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\): \(\frac{4}{3} \times \frac{3}{4} x = \frac{4}{3} \times 21\)
03

Simplify the equation

Simplify the left side \(\frac{4}{3} \times \frac{3}{4} x = x\) and the right side \(\frac{4}{3} \times 21 = 28\). Thus, \(x = 28\).
04

Verify the solution

Substitute \(x = 28\) back into the original equation to check the solution: \(\frac{3}{4} \times 28 = 21\). Since both sides equal, the solution is verified.

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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 linear equations, one of the first steps is to isolate the variable. This means you want to get the variable (in this case, x) by itself on one side of the equation. For the equation \(\frac{3}{4} x = 21\), we need to get rid of the fraction \(\frac{3}{4}\) that is attached to x. To do this, we will multiply both sides of the equation by the reciprocal of \(\frac{3}{4}\).
The reciprocal is simply the flipped version of the fraction. So, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). Multiplying both sides of the equation by \(\frac{4}{3}\) will help to isolate x.
reciprocal
The reciprocal of a number or fraction is what you multiply by to get a product of 1. For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\) because \(\frac{3}{4} \times \frac{4}{3} = 1\). Understanding reciprocals is very important when solving equations because it helps you remove fractions that are attached to your variable.
In our example, when we multiply both sides of the equation \(\frac{3}{4}x = 21\) by \(\frac{4}{3}\), we get:
\[ \frac{4}{3} \times \frac{3}{4} x = \frac{4}{3} \times 21 \]
The left side simplifies to x and on the right side, we multiply \(\frac{4}{3}\) by 21, which simplifies to 28.
Thus, we find that \ x = 28 \.
verify the solution
After finding the solution to an equation, the last step is to verify that the solution is correct. This ensures that there were no mistakes during the calculation. To verify, substitute the value of the variable back into the original equation and check if both sides are equal.
For our solution \(\frac{3}{4} \times 28 = 21\), when we substitute x with 28, we get:
\[ \frac{3}{4} \times 28 = 21 \]
Simplifying the left-hand side, we find:
\[ 21 = 21 \]
This confirms our solution because both sides of the equation are equal, verifying that x = 28 is indeed the correct solution.

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