Problem 78 - Algebra A Combined Function

Question

Decide whether the given number is a solution of the given equation. $$ \frac{2}{7} x=\frac{3}{14} ; 6 $$

Step-by-Step Solution

Answer

No, 6 is not a solution of the equation.

Step 1: Substitute the Given Number into the Equation

To determine if 6 is a solution to the equation \( \frac{2}{7}x = \frac{3}{14} \), substitute \( x = 6 \) into the equation. This gives us \( \frac{2}{7} \times 6 \).

Step 2: Perform Multiplication

Calculate the left side of the equation after substitution: \( \frac{2}{7} \times 6 = \frac{2 \times 6}{7} = \frac{12}{7} \).

Step 3: Compare Both Sides of the Equation

Determine if \( \frac{12}{7} \) is equal to \( \frac{3}{14} \). To compare, convert \( \frac{3}{14} \) to a denominator of 7 or convert \( \frac{12}{7} \) to a denominator of 14. Let's convert \( \frac{12}{7} \) to \( \frac{24}{14} \).

Step 4: Assess Equality

Upon conversion, we see that \( \frac{24}{14} eq \frac{3}{14} \). Therefore, the left side does not equal the right side of the equation.

Key Concepts

Substitution Method

The substitution method is an effective technique used to determine whether a particular value is the solution to an equation. In this approach, you replace the variable in the equation with the given number and simplify the expression to see if the statement holds true. For example, in our exercise, we substitute the number 6 for the variable \( x \) in the equation \( \frac{2}{7} x = \frac{3}{14} \). By doing this, the equation becomes \( \frac{2}{7} \times 6 \). The goal is to check if the expression on the left-hand side equals the fraction on the right-hand side after simplification. If they are equal, the number is a solution; otherwise, it is not.

• Substitute the number into the equation where the variable appears.
• Simplify the expression using arithmetic operations.
• Compare both sides to see if they are equal.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value or proportion. They are useful when comparing or performing operations with fractions. To identify equivalent fractions, you can multiply or divide the numerator and the denominator of a fraction by the same number. This keeps the fraction's value unchanged. In our problem, we convert \( \frac{12}{7} \) to an equivalent fraction with a denominator of 14 by multiplying both the numerator and denominator by 2. This gives us \( \frac{24}{14} \), which can then be compared to \( \frac{3}{14} \).

• Multiply or divide both the numerator and denominator by the same number.
• Ensure the fractions have a common denominator for easy comparison.
• Check if they represent the same proportion or value.

Multiplication of Fractions

Multiplying fractions involves a straightforward process of multiplying the numerators together and the denominators together. This results in a new fraction. For instance, when we replaced \( x \) with 6 in our equation \( \frac{2}{7} \times 6 \), we essentially rewrote 6 as \( \frac{6}{1} \), then multiplied the numerators \( 2 \times 6 = 12 \) and the denominators \( 7 \times 1 = 7 \). Hence, the product is \( \frac{12}{7} \).

• Rewrite whole numbers as fractions (e.g., 6 becomes \( \frac{6}{1}\)).
• Multiply the numerators to find the new numerator.
• Multiply the denominators to find the new denominator.
• Simplify the fraction if possible.