Problem 38 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{x-1}{7}=\frac{2}{21} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{5}{3} \).

Step 1: Cross Multiply

To solve the proportion \( \frac{x-1}{7} = \frac{2}{21} \), begin by cross-multiplying the terms. This means multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This results in the equation:\[(x-1) \times 21 = 2 \times 7\]

Step 2: Calculate Products

Next, calculate the products from step 1.\[21(x-1) = 14\]Now, simplify the expression by distributing the multiplication over the subtraction:\[21x - 21 = 14\]

Step 3: Isolate the Variable

To isolate \( x \), first add 21 to both sides to eliminate the \(-21\) next to \(21x\):\[21x = 35\]

Step 4: Solve for x

Finally, divide both sides of the equation by 21 to solve for \( x \):\[x = \frac{35}{21}\]Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, 7:\[x = \frac{5}{3}\]

Key Concepts

Cross Multiplication

Cross multiplication is a powerful technique used to solve proportions, which are equations that express that two ratios are equal. Imagine you have two fractions set equal to one another, like \( \frac{a}{b} = \frac{c}{d} \). To solve for a variable using cross multiplication, follow these steps:

• Multiply the numerator of the first fraction by the denominator of the second fraction: \( a \times d \).
• Multiply the numerator of the second fraction by the denominator of the first fraction: \( b \times c \).

This technique gives you the equation \( a \times d = b \times c \). The main benefit of cross multiplication is that it eliminates the fractions, making the equation easier to solve.

For our exercise, the proportion \( \frac{x-1}{7} = \frac{2}{21} \) is solved by cross multiplying, resulting in \((x-1) \times 21 = 2 \times 7\). This step essentially removes the fractions, giving us a clear equation to work with.

Variables

In mathematics, a variable is a symbol used to represent an unknown value. Typically, letters like \(x\), \(y\), or \(z\) are used as variables. These symbols can stand for numbers that vary or numbers we are trying to find.

In our example, \(x\) is the variable, and our goal is to determine its value that makes the equation true. When dealing with variables:

• Understand that they hold the place of unknown numbers.
• Work to isolate and solve for them, as they are key to unlocking the solution.

With a little patience and practice, manipulating variables becomes second nature. In our exercise, \(x-1\) is part of the fraction; hence, solving for \(x\) helps resolve the equation.

Isolating Variables

Isolating the variable means getting the variable by itself on one side of the equation. This is crucial for finding its value. Here’s how you can isolate a variable step-by-step:

• First, perform any necessary operations to move constant terms to the opposite side of the variable.
• Next, if there's a coefficient (a number multiplied by the variable), divide both sides by this coefficient to solve for the variable.

Using our example, we had \(21x - 21 = 14\). To isolate \(x\), we first added 21 to both sides, leading to \(21x = 35\). Then we divided both sides by 21, resulting in \(x = \frac{35}{21}\). Isolating the variable is an essential skill when solving equations, making the process logical and systematic.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form, which means expressing it using the smallest possible whole numbers. To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In the final step of our problem, \(x = \frac{35}{21}\), we simplified the fraction by dividing both the numerator (35) and the denominator (21) by their GCD, which is 7, resulting in \(x = \frac{5}{3}\). Simplifying fractions helps in portraying an answer in its cleanest form, which is often required in mathematical solutions to ensure accuracy and clarity.