Problem 34 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{3}{6}=\frac{x}{8} $$

Step-by-Step Solution

Answer

The value of \( x \) is 4.

Step 1 - Set up the equation

To solve the proportion \( \frac{3}{6} = \frac{x}{8} \), we will use the property of cross-multiplication. This property states that the cross products of the proportions are equal. In other words, we can write it as: \( 3 \times 8 = 6 \times x \).

Step 2 - Cross-multiply

Apply cross-multiplication to get: \( 3 \times 8 = 6 \times x \). Simplifying the multiplication on the left side gives us: \( 24 = 6x \).

Step 3 - Solve for x

With the equation \( 24 = 6x \), solve for \( x \) by dividing both sides of the equation by 6. This gives: \( x = \frac{24}{6} \).

Step 4 - Simplify

Now, simplify the expression to find the value of \( x \): \( x = 4 \).

Key Concepts

Cross-multiplication

Cross-multiplication is an invaluable technique for solving proportions. A proportion is an equation that shows two ratios are equal, like \( \frac{3}{6} = \frac{x}{8} \). To solve such equations, we use cross-multiplication, which involves multiplying across the equal sign in a diagonal manner. This technique works because if \( \frac{a}{b} = \frac{c}{d} \), then \( a \times d = b \times c \). These cross products are equal, allowing us to transform the equation into a simpler form that is easier to solve. In our example, multiplying 3 and 8 gives the same result as multiplying 6 and \( x \). Whether you're working with whole numbers or variables, cross-multiplication remains consistent. This straightforward process allows you to compare and solve equations involving fractions efficiently.
Remember, use this method whenever you are faced with proportions in math problems.

Solving equations

Once you have cross-multiplied the proportion, you end up with a simpler form of equation that is easier to solve. For instance, in our initial problem \( 3 \times 8 = 6 \times x \), the cross-multiplication gives us the equation \( 24 = 6x \). Solving such equations involves isolating the variable, which means getting \( x \) on one side of the equation by itself.To do this, you need to perform the same mathematical operation on both sides of the equation. Here, we divide both sides by 6 to simplify and solve for \( x \):

• Divide both sides by 6: \( \frac{24}{6} = \frac{6x}{6} \)
• Simplify: \( 4 = x \)

By following these steps, the proportion is solved, and you have found that \( x = 4 \). This shows the importance of operations like division in simplifying equations and finding the desired variables quickly and efficiently.

Algebraic expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators. When solving proportions, you'll often encounter such expressions involving a variable, like \( \frac{x}{8} \).In our equation, the algebraic expression came from translating the statement of proportion into a format suitable for cross-multiplication: \( \frac{3}{6} = \frac{x}{8} \). By understanding how to manipulate these expressions using algebraic rules and operations, such as multiplying and dividing, you can solve for unknown variables. Here are a few basics to keep in mind:

• Variables like \( x \) serve as placeholders for numbers we are trying to determine.
• Operations (addition, subtraction, multiplication, division) are used to transform equations to isolate variables.
• Simplifying expressions is key to reducing the problem to its essentials and making calculations straightforward.

Properties of algebraic expressions allow us to rearrange and interpret equations, making complex problems more manageable.