Problem 77 - Algebra A Combined Function

Question

Decide whether the given number is a solution of the given equation. $$ \frac{1}{3} x=9 ; 27 $$

Step-by-Step Solution

Answer

Yes, 27 is a solution of the equation.

Step 1: Substitute the given number

Substitute the given number 27 into the equation \( \frac{1}{3} x = 9 \) in place of \( x \). This gives us: \( \frac{1}{3} \times 27 \).

Step 2: Perform the multiplication

Calculate \( \frac{1}{3} \times 27 \). To do this, multiply 27 by 1, which gives 27, and then divide by 3, resulting in 9.

Step 3: Compare the result

The result from Step 2 is 9. Compare this result to the right side of the equation, which is also 9. Since both sides are equal, 27 is a solution to the equation.

Key Concepts

Substitution Method

The substitution method is a technique used to solve equations by replacing variables with specific values. In our example, the equation is \( \frac{1}{3} x = 9 \) and we want to verify whether 27 is the correct solution.

**How to Substitute:**

• Start by identifying the variable in your equation which in this case is \( x \).
• Replace this variable with the number you suspect to be the solution. Here, substitute 27 in place of \( x \).
• The equation becomes \( \frac{1}{3} \times 27 = 9 \).

This substitution allows you to test whether your guessed number satisfies the equation. It's a simple and effective method to check your answers.

Basic Algebra

Basic algebra involves the study of mathematical symbols and the rules for manipulating these symbols. It is the foundation of solving equations like \( \frac{1}{3} x = 9 \).

**Understanding the Equation:**

• The equation \( \frac{1}{3} x = 9 \) involves solving for \( x \) by isolating this variable.
• This usually requires performing operations that will maintain the balance of the equation.
• By substituting 27 into the equation and simplifying, we affirm that both sides are equal.

Mastering basic algebraic techniques helps in simplifying equations and finding the correct solutions efficiently.

Multiplication of Fractions

Multiplication of fractions is a key concept in solving equations involving fractional coefficients, such as \( \frac{1}{3} x = 9 \). Understanding how to perform multiplication with fractions is crucial.

**Steps in Multiplying Fractions:**

• To multiply a fraction by a number, first multiply the numerator by that number. Here, \( 1 \times 27 = 27 \).
• Next, divide the result by the denominator of the fraction: \( 27 \div 3 = 9 \).
• Write down this simplified result.

This operation transforms complex-looking equations into simpler forms, making it easier to verify if a number is a solution. By fully grasping multiplication of fractions, solving similar algebraic problems becomes less daunting.