Problem 32 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 3. $$ \frac{9}{7} d=81 $$

Step-by-Step Solution

Answer

The solution is \(d = 63\).

Step 1: Isolate the Variable

To solve for \(d\), we first need to isolate it on one side of the equation. The equation given is \( \frac{9}{7} d = 81 \). We do this by multiplying both sides by the reciprocal of \( \frac{9}{7} \), which is \( \frac{7}{9} \). Thus, the equation becomes:\[ d = 81 \times \frac{7}{9} \]

Step 2: Simplify the Expression

Multiply 81 by \( \frac{7}{9} \) to solve for \(d\). First, calculate \( 81 \div 9 \), which is 9. Then, multiply the result by 7:\[ d = 9 \times 7 = 63 \]

Step 3: Verify the Solution

To verify the solution, substitute \(d = 63\) back into the original equation to check if both sides are equal:\[ \frac{9}{7} \times 63 = 81 \]Simplify the left side:\[ 9 \times 9 = 81 \]Since the left side equals the right side, \(d=63\) is indeed the correct solution.

Key Concepts

Isolating Variables

In solving equations, one of the first tasks is to isolate the variable, making it easier to find its value. In our original equation, \( \frac{9}{7}d = 81 \), we want to rearrange the formula so that \( d \) stands alone on one side. This is what isolating the variable means.To achieve this, we use operations that will cancel out other numbers or terms from one side. Here, \( d \) is being multiplied by \( \frac{9}{7} \). To isolate \( d \), we want to "undo" this multiplication by \( \frac{9}{7} \). The simplest way to do this is by multiplying both sides of the equation by the reciprocal of \( \frac{9}{7} \).The reciprocal of a fraction is simply flipping the numerator and denominator. Therefore, the reciprocal of \( \frac{9}{7} \) is \( \frac{7}{9} \). By multiplying both sides of the equation by \( \frac{7}{9} \), we effectively neutralize the fraction on the left side, leaving us with just \( d \) on that side:

• Multiply the whole equation by \( \frac{7}{9} \)
• The equation then becomes \( d = 81 \times \frac{7}{9} \)

This clears our path to find out what \( d \) equals, making step-by-step solving much more straightforward.

Reciprocal Multiplication

Reciprocal multiplication is a powerful tool in algebra for solving equations where the variable is part of a fraction. When you multiply a fraction by its reciprocal, the result is always 1. This property is extremely useful because it allows us to eliminate fractions in equations, making calculations simpler.In the equation \( \frac{9}{7}d = 81 \), the reciprocal of \( \frac{9}{7} \) is \( \frac{7}{9} \). Multiplying both sides of the equation by this reciprocal effectively gets rid of the fraction:

• When you multiply: \( \frac{9}{7} \times \frac{7}{9} = 1 \)
• This leaves \( d \) alone on one side of the equation
• Thus, we find: \( d = 81 \times \frac{7}{9} \)

The right side of the equation then requires simple multiplication and division:

• First, divide: \( 81 \div 9 = 9 \)
• Next, multiply: \( 9 \times 7 = 63 \)

By following these steps, the solution becomes clear and understandable.

Verification of Solutions

After finding a solution to an equation, it's crucial to verify its correctness. This step ensures the steps we followed actually lead to a valid answer. Verification helps confirm the arithmetic has been handled correctly. In our example, the proposed solution is \( d = 63 \). To verify:

• Substitute \( d = 63 \) back into the original equation: \( \frac{9}{7} \times 63 = 81 \)
• Calculate: \( 9 \times 9 = 81 \)

We see that both sides of the equation equal 81, thus confirming that the value of \( d \) is indeed correct.Verification can often seem tedious, but it's an essential part of ensuring reliability in mathematical solutions. It gives you confidence in the results and helps prevent errors in your work. Always take a moment to check your answers, as this solidifies understanding and accuracy of your solutions.