Problem 14 - Algebra 1: Concepts and Skills

Question

Solve the proportion using the reciprocal property. Check your solution. $$ \frac{13}{z}=\frac{1}{3} $$

Step-by-Step Solution

Answer

The solution to the proportion \(\frac{13}{z} = \frac{1}{3}\) is \(z = 39\).

Step 1: Apply the Reciprocal Property

Taking the reciprocal of both sides of the proportion, gives: \(\frac{z}{13} = 3\). This is because the reciprocal of \(\frac{13}{z}\) is \(\frac{z}{13}\) and the reciprocal of \(\frac{1}{3}\) is \(3\).

Step 2: Solve the proportion

The value of \(z\) can be found by multiplying both sides of the equation by \(13\): \((\frac{z}{13})*13 = 3*13\), which simplifies to \(z = 39\).

Step 3: Check the Result

To check the result, substitute \(z = 39\) back into the original proportion: \(\frac{13}{39}= \frac{1}{3}\). Simplify the left side: \(\frac{13}{39}\) becomes \(\frac{1}{3}\). Hence, the left side equals the right side, which means that the solution is correct.

Key Concepts

Reciprocal Property

The reciprocal property of equality is a foundational concept in algebra that allows us to solve proportions conveniently. A reciprocal, simply put, is a flipped version of a fraction or number. For instance, the reciprocal of a fraction like \(\frac{a}{b}\) is \(\frac{b}{a}\) when both a and b are non-zero. When dealing with a number 'n', its reciprocal is \(\frac{1}{n}\). The reciprocal of 1, incidentally, is also 1 since flipping it won't change its value.

Understanding how to use this property is essential, particularly when faced with directly proportional relationships. In the example given, \(\frac{13}{z} = \frac{1}{3}\), by taking the reciprocal of both sides, we effectively flip the equation. This move is valid because the reciprocal property ensures that if two ratios are equal, their reciprocals will also be equal. This provides a pathway to simplify the proportion and make z easily isolatable, facilitating the subsequent solution to the problem.

Proportion Problem Solving

Proportion problem solving involves finding the value of a variable that maintains the equality of two ratios. This task is frequently encountered in algebra and requires methodical algebraic manipulation. In the provided example, we approach the problem in structured steps. After applying the reciprocal property, we multiply both sides of the equation by the denominator of the isolated variable to get rid of the fraction.

This step is like cross-multiplication, which is another common technique in solving proportions - except here we applied the reciprocal property first. By doing this, we ensure that the variable is by itself on one side of the equation, making it straightforward to find its value. Here's a tip: always perform the inverse mathematical operation to both sides of an equation when trying to isolate a variable. In our case, multiplying both sides by 13 cancels out the division and leaves us with the value of z.

Algebraic Reasoning

Algebraic reasoning is the process of applying mathematical logic and properties to manipulate algebraic expressions and equations to reach a solution. It's not just about finding an answer; it's also understanding why methods work and ensuring solutions are valid. Using algebraic reasoning, we reasoned that if \(\frac{13}{z} = \frac{1}{3}\), then z must be a number that makes the proportion true. By applying different properties, such as the reciprocal property, and by checking our work, we use logic and known mathematical concepts to confirm the integrity of our solution.

Checking the solution by plugging the value back into the original equation is a crucial step in algebraic reasoning. It validates the accuracy of our work and our understanding of the concept. Encouraging the habit of double-checking not only solidifies confidence in one's algebraic capabilities but also helps in identifying and learning from any potential mistakes.