Problem 61 - Elementary and Intermediate Algebra

Question

Determine whether the given number is a solution of the given equation. $$ 3 ; \frac{108}{x}=36 $$

Step-by-Step Solution

Answer

3 is a solution of the equation.

Step 1 - Understand the given equation

The equation given is \ \[ \frac{108}{x} = 36 \], and we need to determine if the number 3 is a solution to this equation.

Step 2 - Substitute the number into the equation

Substitute \( x = 3 \) into the equation: \ \[ \frac{108}{3} = 36 \]

Step 3 - Simplify the left-hand side

Compute the left-hand side of the equation: \ \[ \frac{108}{3} = 36 \]

Step 4 - Compare both sides of the equation

Check if both sides of the equation are equal: \ \[ 36 = 36 \]

Step 5 - Conclusion

Since both sides of the equation are equal when \( x = 3 \), 3 is indeed a solution of the equation \[ \frac{108}{x} = 36 \].

Key Concepts

algebraic equations

Algebraic equations form the backbone of many mathematical concepts. An algebraic equation connects two expressions using an equal sign. For example, the equation \[ \frac{108}{x} = 36 \] shows a relationship between 108 and 36 through the variable \( x \). Algebra helps us find unknown variables by creating these kinds of equations. For students, understanding algebraic equations is crucial because they appear in various real-life problems and advanced mathematical topics. The goal is often to solve the equation, meaning we need to find the value of \( x \) that makes the equation true.

substitution method

The substitution method is a powerful tool for solving algebraic equations. This method involves replacing the variable in the equation with a specific number to see if the equation holds true. In our example, we wanted to check if \( x = 3 \) is a solution of the equation \[ \frac{108}{x} = 36 \].

Here's how you can use the substitution method:

• First, replace \( x \) with 3 in the given equation: \[ \frac{108}{3} = 36 \].
• Next, simplify the left-hand side: \[ 108 \div 3 = 36 \].
• Finally, check if both sides are equal: \[ 36 = 36 \].

Since both sides are equal, 3 is indeed a solution to the equation. This method is straightforward and helpful in verifying possible solutions.

solutions of equations

Finding solutions of equations means figuring out the values for the variable that make the equation true. In our equation \[ \frac{108}{x} = 36 \], we aimed to check if 3 is a solution.

To do this, we substituted 3 in place of \( x \) and performed the necessary calculations. By verifying that both sides of the equation were equal, we confirmed that 3 is indeed a solution.

It's essential to follow these steps when solving equations:

• Substitute the proposed solution into the equation.
• Simplify either side (if necessary).
• Compare both sides to check for equality.

If the sides are equal, the number is a solution. This approach helps ensure that the variable value you find is correct.

fractions in equations

Fractions in equations might seem tricky, but they follow the same rules as any other algebraic equations. In our example, we dealt with the fraction \[ \frac{108}{x} = 36 \].

When solving equations involving fractions, consider these simple tips:

• Simplify the fraction (if possible).
• Identify the type of fraction (numerator/denominator).
• Substitute the given number and perform basic arithmetic.

In our case, we substituted \( x = 3 \), which made the equation \[ \frac{108}{3} = 36 \]. Simplification resulted in both sides equaling 36, showing that 3 is a solution.

Remember, with some practice, fractions in equations become much easier to handle. The key is to balance both sides of the equation properly.